LMFDB - Embedded newform 1568.3.d.c.1471.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(1471,1568)]

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, 0, 0]))

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

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

chi := DirichletCharacter("1568.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.d (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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1568.1471
Dual form			1568.3.d.c.1471.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000 q^{5} +8.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000 q^{5} +8.00000 q^{9} -17.0000i q^{11} -24.0000 q^{13} +1.00000i q^{15} -1.00000 q^{17} +7.00000i q^{19} +7.00000i q^{23} -24.0000 q^{25} -17.0000i q^{27} +24.0000 q^{29} +41.0000i q^{31} -17.0000 q^{33} -49.0000 q^{37} +24.0000i q^{39} +48.0000 q^{41} -24.0000i q^{43} -8.00000 q^{45} +55.0000i q^{47} +1.00000i q^{51} -25.0000 q^{53} +17.0000i q^{55} +7.00000 q^{57} -17.0000i q^{59} +1.00000 q^{61} +24.0000 q^{65} +65.0000i q^{67} +7.00000 q^{69} -96.0000i q^{71} -95.0000 q^{73} +24.0000i q^{75} -41.0000i q^{79} +55.0000 q^{81} +72.0000i q^{83} +1.00000 q^{85} -24.0000i q^{87} -95.0000 q^{89} +41.0000 q^{93} -7.00000i q^{95} -144.000 q^{97} -136.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 16 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 16 * q^9	$ 2 q - 2 q^{5} + 16 q^{9} - 48 q^{13} - 2 q^{17} - 48 q^{25} + 48 q^{29} - 34 q^{33} - 98 q^{37} + 96 q^{41} - 16 q^{45} - 50 q^{53} + 14 q^{57} + 2 q^{61} + 48 q^{65} + 14 q^{69} - 190 q^{73} + 110 q^{81} + 2 q^{85} - 190 q^{89} + 82 q^{93} - 288 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 16 * q^9 - 48 * q^13 - 2 * q^17 - 48 * q^25 + 48 * q^29 - 34 * q^33 - 98 * q^37 + 96 * q^41 - 16 * q^45 - 50 * q^53 + 14 * q^57 + 2 * q^61 + 48 * q^65 + 14 * q^69 - 190 * q^73 + 110 * q^81 + 2 * q^85 - 190 * q^89 + 82 * q^93 - 288 * q^97

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
$3$	− 1.00000i	− 0.333333i	−0.986013	−	0.166667i	$-0.946700\pi$
$3$	− 1.00000i	− 0.333333i	0.986013	−	0.166667i	$-0.0533004\pi$
$4$	0	0
$5$	−1.00000	−0.200000	−0.100000	−	0.994987i	$-0.531884\pi$
$5$	−1.00000	−0.200000	−0.100000	+	0.994987i	$0.531884\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.00000	0.888889
$10$	0	0
$11$	− 17.0000i	− 1.54545i	−0.634738	−	0.772727i	$-0.718892\pi$
$11$	− 17.0000i	− 1.54545i	0.634738	−	0.772727i	$-0.281108\pi$
$12$	0	0
$13$	−24.0000	−1.84615	−0.923077	−	0.384615i	$-0.874334\pi$
$13$	−24.0000	−1.84615	−0.923077	+	0.384615i	$0.874334\pi$
$14$	0	0
$15$	1.00000i	0.0666667i
$16$	0	0
$17$	−1.00000	−0.0588235	−0.0294118	−	0.999567i	$-0.509363\pi$
$17$	−1.00000	−0.0588235	−0.0294118	+	0.999567i	$0.509363\pi$
$18$	0	0
$19$	7.00000i	0.368421i	0.982887	+	0.184211i	$0.0589728\pi$
$19$	7.00000i	0.368421i	−0.982887	+	0.184211i	$0.941027\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.00000i	0.304348i	0.988354	+	0.152174i	$0.0486274\pi$
$23$	7.00000i	0.304348i	−0.988354	+	0.152174i	$0.951373\pi$
$24$	0	0
$25$	−24.0000	−0.960000
$26$	0	0
$27$	− 17.0000i	− 0.629630i
$28$	0	0
$29$	24.0000	0.827586	0.413793	−	0.910371i	$-0.364204\pi$
$29$	24.0000	0.827586	0.413793	+	0.910371i	$0.364204\pi$
$30$	0	0
$31$	41.0000i	1.32258i	0.750130	+	0.661290i	$0.229991\pi$
$31$	41.0000i	1.32258i	−0.750130	+	0.661290i	$0.770009\pi$
$32$	0	0
$33$	−17.0000	−0.515152
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−49.0000	−1.32432	−0.662162	−	0.749361i	$-0.730361\pi$
$37$	−49.0000	−1.32432	−0.662162	+	0.749361i	$0.730361\pi$
$38$	0	0
$39$	24.0000i	0.615385i
$40$	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
$43$	− 24.0000i	− 0.558140i	−0.960271	−	0.279070i	$-0.909974\pi$
$43$	− 24.0000i	− 0.558140i	0.960271	−	0.279070i	$-0.0900261\pi$
$44$	0	0
$45$	−8.00000	−0.177778
$46$	0	0
$47$	55.0000i	1.17021i	0.810957	+	0.585106i	$0.198947\pi$
$47$	55.0000i	1.17021i	−0.810957	+	0.585106i	$0.801053\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.00000i	0.0196078i
$52$	0	0
$53$	−25.0000	−0.471698	−0.235849	−	0.971790i	$-0.575787\pi$
$53$	−25.0000	−0.471698	−0.235849	+	0.971790i	$0.575787\pi$
$54$	0	0
$55$	17.0000i	0.309091i
$56$	0	0
$57$	7.00000	0.122807
$58$	0	0
$59$	− 17.0000i	− 0.288136i	−0.989568	−	0.144068i	$-0.953982\pi$
$59$	− 17.0000i	− 0.288136i	0.989568	−	0.144068i	$-0.0460183\pi$
$60$	0	0
$61$	1.00000	0.0163934	0.00819672	−	0.999966i	$-0.497391\pi$
$61$	1.00000	0.0163934	0.00819672	+	0.999966i	$0.497391\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	24.0000	0.369231
$66$	0	0
$67$	65.0000i	0.970149i	0.874473	+	0.485075i	$0.161208\pi$
$67$	65.0000i	0.970149i	−0.874473	+	0.485075i	$0.838792\pi$
$68$	0	0
$69$	7.00000	0.101449
$70$	0	0
$71$	− 96.0000i	− 1.35211i	−0.736850	−	0.676056i	$-0.763688\pi$
$71$	− 96.0000i	− 1.35211i	0.736850	−	0.676056i	$-0.236312\pi$
$72$	0	0
$73$	−95.0000	−1.30137	−0.650685	−	0.759348i	$-0.725518\pi$
$73$	−95.0000	−1.30137	−0.650685	+	0.759348i	$0.725518\pi$
$74$	0	0
$75$	24.0000i	0.320000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 41.0000i	− 0.518987i	−0.965745	−	0.259494i	$-0.916444\pi$
$79$	− 41.0000i	− 0.518987i	0.965745	−	0.259494i	$-0.0835557\pi$
$80$	0	0
$81$	55.0000	0.679012
$82$	0	0
$83$	72.0000i	0.867470i	0.901041	+	0.433735i	$0.142805\pi$
$83$	72.0000i	0.867470i	−0.901041	+	0.433735i	$0.857195\pi$
$84$	0	0
$85$	1.00000	0.0117647
$86$	0	0
$87$	− 24.0000i	− 0.275862i
$88$	0	0
$89$	−95.0000	−1.06742	−0.533708	−	0.845669i	$-0.679202\pi$
$89$	−95.0000	−1.06742	−0.533708	+	0.845669i	$0.679202\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	41.0000	0.440860
$94$	0	0
$95$	− 7.00000i	− 0.0736842i
$96$	0	0
$97$	−144.000	−1.48454	−0.742268	−	0.670103i	$-0.766250\pi$
$97$	−144.000	−1.48454	−0.742268	+	0.670103i	$0.766250\pi$
$98$	0	0
$99$	− 136.000i	− 1.37374i
$100$	0	0
$101$	−73.0000	−0.722772	−0.361386	−	0.932416i	$-0.617696\pi$
$101$	−73.0000	−0.722772	−0.361386	+	0.932416i	$0.617696\pi$
$102$	0	0
$103$	89.0000i	0.864078i	0.901855	+	0.432039i	$0.142206\pi$
$103$	89.0000i	0.864078i	−0.901855	+	0.432039i	$0.857794\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	185.000i	1.72897i	0.502657	+	0.864486i	$0.332356\pi$
$107$	185.000i	1.72897i	−0.502657	+	0.864486i	$0.667644\pi$
$108$	0	0
$109$	−71.0000	−0.651376	−0.325688	−	0.945477i	$-0.605596\pi$
$109$	−71.0000	−0.651376	−0.325688	+	0.945477i	$0.605596\pi$
$110$	0	0
$111$	49.0000i	0.441441i
$112$	0	0
$113$	96.0000	0.849558	0.424779	−	0.905297i	$-0.360352\pi$
$113$	96.0000	0.849558	0.424779	+	0.905297i	$0.360352\pi$
$114$	0	0
$115$	− 7.00000i	− 0.0608696i
$116$	0	0
$117$	−192.000	−1.64103
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−168.000	−1.38843
$122$	0	0
$123$	− 48.0000i	− 0.390244i
$124$	0	0
$125$	49.0000	0.392000
$126$	0	0
$127$	144.000i	1.13386i	0.823767	+	0.566929i	$0.191869\pi$
$127$	144.000i	1.13386i	−0.823767	+	0.566929i	$0.808131\pi$
$128$	0	0
$129$	−24.0000	−0.186047
$130$	0	0
$131$	185.000i	1.41221i	0.708105	+	0.706107i	$0.249550\pi$
$131$	185.000i	1.41221i	−0.708105	+	0.706107i	$0.750450\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	17.0000i	0.125926i
$136$	0	0
$137$	−143.000	−1.04380	−0.521898	−	0.853008i	$-0.674776\pi$
$137$	−143.000	−1.04380	−0.521898	+	0.853008i	$0.674776\pi$
$138$	0	0
$139$	− 216.000i	− 1.55396i	−0.629527	−	0.776978i	$-0.716751\pi$
$139$	− 216.000i	− 1.55396i	0.629527	−	0.776978i	$-0.283249\pi$
$140$	0	0
$141$	55.0000	0.390071
$142$	0	0
$143$	408.000i	2.85315i
$144$	0	0
$145$	−24.0000	−0.165517
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−47.0000	−0.315436	−0.157718	−	0.987484i	$-0.550414\pi$
$149$	−47.0000	−0.315436	−0.157718	+	0.987484i	$0.550414\pi$
$150$	0	0
$151$	199.000i	1.31788i	0.752195	+	0.658940i	$0.228995\pi$
$151$	199.000i	1.31788i	−0.752195	+	0.658940i	$0.771005\pi$
$152$	0	0
$153$	−8.00000	−0.0522876
$154$	0	0
$155$	− 41.0000i	− 0.264516i
$156$	0	0
$157$	73.0000	0.464968	0.232484	−	0.972600i	$-0.425315\pi$
$157$	73.0000	0.464968	0.232484	+	0.972600i	$0.425315\pi$
$158$	0	0
$159$	25.0000i	0.157233i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	55.0000i	0.337423i	0.985665	+	0.168712i	$0.0539607\pi$
$163$	55.0000i	0.337423i	−0.985665	+	0.168712i	$0.946039\pi$
$164$	0	0
$165$	17.0000	0.103030
$166$	0	0
$167$	206.000i	1.23353i	0.787146	+	0.616766i	$0.211558\pi$
$167$	206.000i	1.23353i	−0.787146	+	0.616766i	$0.788442\pi$
$168$	0	0
$169$	407.000	2.40828
$170$	0	0
$171$	56.0000i	0.327485i
$172$	0	0
$173$	−239.000	−1.38150	−0.690751	−	0.723092i	$-0.742720\pi$
$173$	−239.000	−1.38150	−0.690751	+	0.723092i	$0.742720\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−17.0000	−0.0960452
$178$	0	0
$179$	− 17.0000i	− 0.0949721i	−0.998872	−	0.0474860i	$-0.984879\pi$
$179$	− 17.0000i	− 0.0949721i	0.998872	−	0.0474860i	$-0.0151210\pi$
$180$	0	0
$181$	−70.0000	−0.386740	−0.193370	−	0.981126i	$-0.561942\pi$
$181$	−70.0000	−0.386740	−0.193370	+	0.981126i	$0.561942\pi$
$182$	0	0
$183$	− 1.00000i	− 0.00546448i
$184$	0	0
$185$	49.0000	0.264865
$186$	0	0
$187$	17.0000i	0.0909091i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 199.000i	− 1.04188i	−0.853592	−	0.520942i	$-0.825581\pi$
$191$	− 199.000i	− 1.04188i	0.853592	−	0.520942i	$-0.174419\pi$
$192$	0	0
$193$	−47.0000	−0.243523	−0.121762	−	0.992559i	$-0.538854\pi$
$193$	−47.0000	−0.243523	−0.121762	+	0.992559i	$0.538854\pi$
$194$	0	0
$195$	− 24.0000i	− 0.123077i
$196$	0	0
$197$	24.0000	0.121827	0.0609137	−	0.998143i	$-0.480599\pi$
$197$	24.0000	0.121827	0.0609137	+	0.998143i	$0.480599\pi$
$198$	0	0
$199$	137.000i	0.688442i	0.938889	+	0.344221i	$0.111857\pi$
$199$	137.000i	0.688442i	−0.938889	+	0.344221i	$0.888143\pi$
$200$	0	0
$201$	65.0000	0.323383
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−48.0000	−0.234146
$206$	0	0
$207$	56.0000i	0.270531i
$208$	0	0
$209$	119.000	0.569378
$210$	0	0
$211$	− 264.000i	− 1.25118i	−0.780150	−	0.625592i	$-0.784857\pi$
$211$	− 264.000i	− 1.25118i	0.780150	−	0.625592i	$-0.215143\pi$
$212$	0	0
$213$	−96.0000	−0.450704
$214$	0	0
$215$	24.0000i	0.111628i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	95.0000i	0.433790i
$220$	0	0
$221$	24.0000	0.108597
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−192.000	−0.853333
$226$	0	0
$227$	− 415.000i	− 1.82819i	−0.405496	−	0.914097i	$-0.632901\pi$
$227$	− 415.000i	− 1.82819i	0.405496	−	0.914097i	$-0.367099\pi$
$228$	0	0
$229$	143.000	0.624454	0.312227	−	0.950008i	$-0.398925\pi$
$229$	143.000	0.624454	0.312227	+	0.950008i	$0.398925\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	289.000	1.24034	0.620172	−	0.784466i	$-0.287063\pi$
$233$	289.000	1.24034	0.620172	+	0.784466i	$0.287063\pi$
$234$	0	0
$235$	− 55.0000i	− 0.234043i
$236$	0	0
$237$	−41.0000	−0.172996
$238$	0	0
$239$	− 226.000i	− 0.945607i	−0.881168	−	0.472803i	$-0.843242\pi$
$239$	− 226.000i	− 0.945607i	0.881168	−	0.472803i	$-0.156758\pi$
$240$	0	0
$241$	−95.0000	−0.394191	−0.197095	−	0.980384i	$-0.563151\pi$
$241$	−95.0000	−0.394191	−0.197095	+	0.980384i	$0.563151\pi$
$242$	0	0
$243$	− 208.000i	− 0.855967i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 168.000i	− 0.680162i
$248$	0	0
$249$	72.0000	0.289157
$250$	0	0
$251$	− 38.0000i	− 0.151394i	−0.997131	−	0.0756972i	$-0.975882\pi$
$251$	− 38.0000i	− 0.151394i	0.997131	−	0.0756972i	$-0.0241182\pi$
$252$	0	0
$253$	119.000	0.470356
$254$	0	0
$255$	− 1.00000i	− 0.00392157i
$256$	0	0
$257$	−193.000	−0.750973	−0.375486	−	0.926828i	$-0.622524\pi$
$257$	−193.000	−0.750973	−0.375486	+	0.926828i	$0.622524\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	192.000	0.735632
$262$	0	0
$263$	233.000i	0.885932i	0.896539	+	0.442966i	$0.146074\pi$
$263$	233.000i	0.885932i	−0.896539	+	0.442966i	$0.853926\pi$
$264$	0	0
$265$	25.0000	0.0943396
$266$	0	0
$267$	95.0000i	0.355805i
$268$	0	0
$269$	−455.000	−1.69145	−0.845725	−	0.533619i	$-0.820832\pi$
$269$	−455.000	−1.69145	−0.845725	+	0.533619i	$0.820832\pi$
$270$	0	0
$271$	425.000i	1.56827i	0.620593	+	0.784133i	$0.286892\pi$
$271$	425.000i	1.56827i	−0.620593	+	0.784133i	$0.713108\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	408.000i	1.48364i
$276$	0	0
$277$	167.000	0.602888	0.301444	−	0.953484i	$-0.402531\pi$
$277$	167.000	0.602888	0.301444	+	0.953484i	$0.402531\pi$
$278$	0	0
$279$	328.000i	1.17563i
$280$	0	0
$281$	−432.000	−1.53737	−0.768683	−	0.639630i	$-0.779088\pi$
$281$	−432.000	−1.53737	−0.768683	+	0.639630i	$0.779088\pi$
$282$	0	0
$283$	− 223.000i	− 0.787986i	−0.919113	−	0.393993i	$-0.871093\pi$
$283$	− 223.000i	− 0.787986i	0.919113	−	0.393993i	$-0.128907\pi$
$284$	0	0
$285$	−7.00000	−0.0245614
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−288.000	−0.996540
$290$	0	0
$291$	144.000i	0.494845i
$292$	0	0
$293$	−26.0000	−0.0887372	−0.0443686	−	0.999015i	$-0.514128\pi$
$293$	−26.0000	−0.0887372	−0.0443686	+	0.999015i	$0.514128\pi$
$294$	0	0
$295$	17.0000i	0.0576271i
$296$	0	0
$297$	−289.000	−0.973064
$298$	0	0
$299$	− 168.000i	− 0.561873i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	73.0000i	0.240924i
$304$	0	0
$305$	−1.00000	−0.00327869
$306$	0	0
$307$	− 264.000i	− 0.859935i	−0.902844	−	0.429967i	$-0.858525\pi$
$307$	− 264.000i	− 0.859935i	0.902844	−	0.429967i	$-0.141475\pi$
$308$	0	0
$309$	89.0000	0.288026
$310$	0	0
$311$	343.000i	1.10289i	0.834210	+	0.551447i	$0.185924\pi$
$311$	343.000i	1.10289i	−0.834210	+	0.551447i	$0.814076\pi$
$312$	0	0
$313$	−335.000	−1.07029	−0.535144	−	0.844761i	$-0.679743\pi$
$313$	−335.000	−1.07029	−0.535144	+	0.844761i	$0.679743\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−529.000	−1.66877	−0.834385	−	0.551182i	$-0.814177\pi$
$317$	−529.000	−1.66877	−0.834385	+	0.551182i	$0.814177\pi$
$318$	0	0
$319$	− 408.000i	− 1.27900i
$320$	0	0
$321$	185.000	0.576324
$322$	0	0
$323$	− 7.00000i	− 0.0216718i
$324$	0	0
$325$	576.000	1.77231
$326$	0	0
$327$	71.0000i	0.217125i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	55.0000i	0.166163i	0.996543	+	0.0830816i	$0.0264762\pi$
$331$	55.0000i	0.166163i	−0.996543	+	0.0830816i	$0.973524\pi$
$332$	0	0
$333$	−392.000	−1.17718
$334$	0	0
$335$	− 65.0000i	− 0.194030i
$336$	0	0
$337$	−240.000	−0.712166	−0.356083	−	0.934454i	$-0.615888\pi$
$337$	−240.000	−0.712166	−0.356083	+	0.934454i	$0.615888\pi$
$338$	0	0
$339$	− 96.0000i	− 0.283186i
$340$	0	0
$341$	697.000	2.04399
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.00000	−0.0202899
$346$	0	0
$347$	− 31.0000i	− 0.0893372i	−0.999002	−	0.0446686i	$-0.985777\pi$
$347$	− 31.0000i	− 0.0893372i	0.999002	−	0.0446686i	$-0.0142232\pi$
$348$	0	0
$349$	120.000	0.343840	0.171920	−	0.985111i	$-0.445003\pi$
$349$	120.000	0.343840	0.171920	+	0.985111i	$0.445003\pi$
$350$	0	0
$351$	408.000i	1.16239i
$352$	0	0
$353$	527.000	1.49292	0.746459	−	0.665431i	$-0.231752\pi$
$353$	527.000	1.49292	0.746459	+	0.665431i	$0.231752\pi$
$354$	0	0
$355$	96.0000i	0.270423i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 535.000i	− 1.49025i	−0.666924	−	0.745125i	$-0.732390\pi$
$359$	− 535.000i	− 1.49025i	0.666924	−	0.745125i	$-0.267610\pi$
$360$	0	0
$361$	312.000	0.864266
$362$	0	0
$363$	168.000i	0.462810i
$364$	0	0
$365$	95.0000	0.260274
$366$	0	0
$367$	− 89.0000i	− 0.242507i	−0.992622	−	0.121253i	$-0.961309\pi$
$367$	− 89.0000i	− 0.242507i	0.992622	−	0.121253i	$-0.0386914\pi$
$368$	0	0
$369$	384.000	1.04065
$370$	0	0
$371$	0	0
$372$	0	0
$373$	335.000	0.898123	0.449062	−	0.893501i	$-0.351758\pi$
$373$	335.000	0.898123	0.449062	+	0.893501i	$0.351758\pi$
$374$	0	0
$375$	− 49.0000i	− 0.130667i
$376$	0	0
$377$	−576.000	−1.52785
$378$	0	0
$379$	− 38.0000i	− 0.100264i	−0.998743	−	0.0501319i	$-0.984036\pi$
$379$	− 38.0000i	− 0.100264i	0.998743	−	0.0501319i	$-0.0159642\pi$
$380$	0	0
$381$	144.000	0.377953
$382$	0	0
$383$	− 665.000i	− 1.73629i	−0.496309	−	0.868146i	$-0.665312\pi$
$383$	− 665.000i	− 1.73629i	0.496309	−	0.868146i	$-0.334688\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 192.000i	− 0.496124i
$388$	0	0
$389$	73.0000	0.187661	0.0938303	−	0.995588i	$-0.470089\pi$
$389$	73.0000	0.187661	0.0938303	+	0.995588i	$0.470089\pi$
$390$	0	0
$391$	− 7.00000i	− 0.0179028i
$392$	0	0
$393$	185.000	0.470738
$394$	0	0
$395$	41.0000i	0.103797i
$396$	0	0
$397$	−1.00000	−0.00251889	−0.00125945	−	0.999999i	$-0.500401\pi$
$397$	−1.00000	−0.00251889	−0.00125945	+	0.999999i	$0.500401\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	337.000	0.840399	0.420200	−	0.907432i	$-0.361960\pi$
$401$	337.000	0.840399	0.420200	+	0.907432i	$0.361960\pi$
$402$	0	0
$403$	− 984.000i	− 2.44169i
$404$	0	0
$405$	−55.0000	−0.135802
$406$	0	0
$407$	833.000i	2.04668i
$408$	0	0
$409$	479.000	1.17115	0.585575	−	0.810619i	$-0.300869\pi$
$409$	479.000	1.17115	0.585575	+	0.810619i	$0.300869\pi$
$410$	0	0
$411$	143.000i	0.347932i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 72.0000i	− 0.173494i
$416$	0	0
$417$	−216.000	−0.517986
$418$	0	0
$419$	− 552.000i	− 1.31742i	−0.752396	−	0.658711i	$-0.771102\pi$
$419$	− 552.000i	− 1.31742i	0.752396	−	0.658711i	$-0.228898\pi$
$420$	0	0
$421$	−216.000	−0.513064	−0.256532	−	0.966536i	$-0.582580\pi$
$421$	−216.000	−0.513064	−0.256532	+	0.966536i	$0.582580\pi$
$422$	0	0
$423$	440.000i	1.04019i
$424$	0	0
$425$	24.0000	0.0564706
$426$	0	0
$427$	0	0
$428$	0	0
$429$	408.000	0.951049
$430$	0	0
$431$	439.000i	1.01856i	0.860600	+	0.509281i	$0.170089\pi$
$431$	439.000i	1.01856i	−0.860600	+	0.509281i	$0.829911\pi$
$432$	0	0
$433$	288.000	0.665127	0.332564	−	0.943081i	$-0.392086\pi$
$433$	288.000	0.665127	0.332564	+	0.943081i	$0.392086\pi$
$434$	0	0
$435$	24.0000i	0.0551724i
$436$	0	0
$437$	−49.0000	−0.112128
$438$	0	0
$439$	− 617.000i	− 1.40547i	−0.711453	−	0.702733i	$-0.751963\pi$
$439$	− 617.000i	− 1.40547i	0.711453	−	0.702733i	$-0.248037\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 247.000i	− 0.557562i	−0.960355	−	0.278781i	$-0.910070\pi$
$443$	− 247.000i	− 0.557562i	0.960355	−	0.278781i	$-0.0899304\pi$
$444$	0	0
$445$	95.0000	0.213483
$446$	0	0
$447$	47.0000i	0.105145i
$448$	0	0
$449$	−288.000	−0.641425	−0.320713	−	0.947177i	$-0.603922\pi$
$449$	−288.000	−0.641425	−0.320713	+	0.947177i	$0.603922\pi$
$450$	0	0
$451$	− 816.000i	− 1.80931i
$452$	0	0
$453$	199.000	0.439294
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−97.0000	−0.212254	−0.106127	−	0.994353i	$-0.533845\pi$
$457$	−97.0000	−0.212254	−0.106127	+	0.994353i	$0.533845\pi$
$458$	0	0
$459$	17.0000i	0.0370370i
$460$	0	0
$461$	−312.000	−0.676790	−0.338395	−	0.941004i	$-0.609884\pi$
$461$	−312.000	−0.676790	−0.338395	+	0.941004i	$0.609884\pi$
$462$	0	0
$463$	− 192.000i	− 0.414687i	−0.978268	−	0.207343i	$-0.933518\pi$
$463$	− 192.000i	− 0.414687i	0.978268	−	0.207343i	$-0.0664817\pi$
$464$	0	0
$465$	−41.0000	−0.0881720
$466$	0	0
$467$	− 295.000i	− 0.631692i	−0.948811	−	0.315846i	$-0.897712\pi$
$467$	− 295.000i	− 0.631692i	0.948811	−	0.315846i	$-0.102288\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 73.0000i	− 0.154989i
$472$	0	0
$473$	−408.000	−0.862579
$474$	0	0
$475$	− 168.000i	− 0.353684i
$476$	0	0
$477$	−200.000	−0.419287
$478$	0	0
$479$	− 617.000i	− 1.28810i	−0.764983	−	0.644050i	$-0.777253\pi$
$479$	− 617.000i	− 1.28810i	0.764983	−	0.644050i	$-0.222747\pi$
$480$	0	0
$481$	1176.00	2.44491
$482$	0	0
$483$	0	0
$484$	0	0
$485$	144.000	0.296907
$486$	0	0
$487$	− 247.000i	− 0.507187i	−0.967311	−	0.253593i	$-0.918387\pi$
$487$	− 247.000i	− 0.507187i	0.967311	−	0.253593i	$-0.0816125\pi$
$488$	0	0
$489$	55.0000	0.112474
$490$	0	0
$491$	− 134.000i	− 0.272912i	−0.990646	−	0.136456i	$-0.956429\pi$
$491$	− 134.000i	− 0.272912i	0.990646	−	0.136456i	$-0.0435713\pi$
$492$	0	0
$493$	−24.0000	−0.0486815
$494$	0	0
$495$	136.000i	0.274747i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	727.000i	1.45691i	0.685092	+	0.728457i	$0.259762\pi$
$499$	727.000i	1.45691i	−0.685092	+	0.728457i	$0.740238\pi$
$500$	0	0
$501$	206.000	0.411178
$502$	0	0
$503$	− 432.000i	− 0.858847i	−0.903103	−	0.429423i	$-0.858717\pi$
$503$	− 432.000i	− 0.858847i	0.903103	−	0.429423i	$-0.141283\pi$
$504$	0	0
$505$	73.0000	0.144554
$506$	0	0
$507$	− 407.000i	− 0.802761i
$508$	0	0
$509$	−433.000	−0.850688	−0.425344	−	0.905032i	$-0.639847\pi$
$509$	−433.000	−0.850688	−0.425344	+	0.905032i	$0.639847\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	119.000	0.231969
$514$	0	0
$515$	− 89.0000i	− 0.172816i
$516$	0	0
$517$	935.000	1.80851
$518$	0	0
$519$	239.000i	0.460501i
$520$	0	0
$521$	−479.000	−0.919386	−0.459693	−	0.888078i	$-0.652041\pi$
$521$	−479.000	−0.919386	−0.459693	+	0.888078i	$0.652041\pi$
$522$	0	0
$523$	− 905.000i	− 1.73040i	−0.501426	−	0.865201i	$-0.667191\pi$
$523$	− 905.000i	− 1.73040i	0.501426	−	0.865201i	$-0.332809\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 41.0000i	− 0.0777989i
$528$	0	0
$529$	480.000	0.907372
$530$	0	0
$531$	− 136.000i	− 0.256121i
$532$	0	0
$533$	−1152.00	−2.16135
$534$	0	0
$535$	− 185.000i	− 0.345794i
$536$	0	0
$537$	−17.0000	−0.0316574
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−191.000	−0.353050	−0.176525	−	0.984296i	$-0.556486\pi$
$541$	−191.000	−0.353050	−0.176525	+	0.984296i	$0.556486\pi$
$542$	0	0
$543$	70.0000i	0.128913i
$544$	0	0
$545$	71.0000	0.130275
$546$	0	0
$547$	374.000i	0.683729i	0.939749	+	0.341865i	$0.111058\pi$
$547$	374.000i	0.683729i	−0.939749	+	0.341865i	$0.888942\pi$
$548$	0	0
$549$	8.00000	0.0145719
$550$	0	0
$551$	168.000i	0.304900i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 49.0000i	− 0.0882883i
$556$	0	0
$557$	937.000	1.68223	0.841113	−	0.540859i	$-0.181901\pi$
$557$	937.000	1.68223	0.841113	+	0.540859i	$0.181901\pi$
$558$	0	0
$559$	576.000i	1.03041i
$560$	0	0
$561$	17.0000	0.0303030
$562$	0	0
$563$	799.000i	1.41918i	0.704613	+	0.709591i	$0.251120\pi$
$563$	799.000i	1.41918i	−0.704613	+	0.709591i	$0.748880\pi$
$564$	0	0
$565$	−96.0000	−0.169912
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−529.000	−0.929701	−0.464851	−	0.885389i	$-0.653892\pi$
$569$	−529.000	−0.929701	−0.464851	+	0.885389i	$0.653892\pi$
$570$	0	0
$571$	1039.00i	1.81961i	0.415031	+	0.909807i	$0.363771\pi$
$571$	1039.00i	1.81961i	−0.415031	+	0.909807i	$0.636229\pi$
$572$	0	0
$573$	−199.000	−0.347295
$574$	0	0
$575$	− 168.000i	− 0.292174i
$576$	0	0
$577$	529.000	0.916811	0.458406	−	0.888743i	$-0.348421\pi$
$577$	529.000	0.916811	0.458406	+	0.888743i	$0.348421\pi$
$578$	0	0
$579$	47.0000i	0.0811744i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	425.000i	0.728988i
$584$	0	0
$585$	192.000	0.328205
$586$	0	0
$587$	840.000i	1.43101i	0.698610	+	0.715503i	$0.253802\pi$
$587$	840.000i	1.43101i	−0.698610	+	0.715503i	$0.746198\pi$
$588$	0	0
$589$	−287.000	−0.487267
$590$	0	0
$591$	− 24.0000i	− 0.0406091i
$592$	0	0
$593$	383.000	0.645868	0.322934	−	0.946421i	$-0.395331\pi$
$593$	383.000	0.645868	0.322934	+	0.946421i	$0.395331\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	137.000	0.229481
$598$	0	0
$599$	− 343.000i	− 0.572621i	−0.958137	−	0.286311i	$-0.907571\pi$
$599$	− 343.000i	− 0.572621i	0.958137	−	0.286311i	$-0.0924289\pi$
$600$	0	0
$601$	624.000	1.03827	0.519135	−	0.854692i	$-0.326254\pi$
$601$	624.000	1.03827	0.519135	+	0.854692i	$0.326254\pi$
$602$	0	0
$603$	520.000i	0.862355i
$604$	0	0
$605$	168.000	0.277686
$606$	0	0
$607$	− 137.000i	− 0.225700i	−0.993612	−	0.112850i	$-0.964002\pi$
$607$	− 137.000i	− 0.225700i	0.993612	−	0.112850i	$-0.0359980\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1320.00i	− 2.16039i
$612$	0	0
$613$	−71.0000	−0.115824	−0.0579119	−	0.998322i	$-0.518444\pi$
$613$	−71.0000	−0.115824	−0.0579119	+	0.998322i	$0.518444\pi$
$614$	0	0
$615$	48.0000i	0.0780488i
$616$	0	0
$617$	384.000	0.622366	0.311183	−	0.950350i	$-0.399275\pi$
$617$	384.000	0.622366	0.311183	+	0.950350i	$0.399275\pi$
$618$	0	0
$619$	− 593.000i	− 0.957997i	−0.877816	−	0.478998i	$-0.841000\pi$
$619$	− 593.000i	− 0.957997i	0.877816	−	0.478998i	$-0.159000\pi$
$620$	0	0
$621$	119.000	0.191626
$622$	0	0
$623$	0	0
$624$	0	0
$625$	551.000	0.881600
$626$	0	0
$627$	− 119.000i	− 0.189793i
$628$	0	0
$629$	49.0000	0.0779014
$630$	0	0
$631$	− 384.000i	− 0.608558i	−0.952583	−	0.304279i	$-0.901585\pi$
$631$	− 384.000i	− 0.608558i	0.952583	−	0.304279i	$-0.0984155\pi$
$632$	0	0
$633$	−264.000	−0.417062
$634$	0	0
$635$	− 144.000i	− 0.226772i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 768.000i	− 1.20188i
$640$	0	0
$641$	767.000	1.19657	0.598284	−	0.801284i	$-0.295849\pi$
$641$	767.000	1.19657	0.598284	+	0.801284i	$0.295849\pi$
$642$	0	0
$643$	456.000i	0.709176i	0.935023	+	0.354588i	$0.115379\pi$
$643$	456.000i	0.709176i	−0.935023	+	0.354588i	$0.884621\pi$
$644$	0	0
$645$	24.0000	0.0372093
$646$	0	0
$647$	− 89.0000i	− 0.137558i	−0.997632	−	0.0687790i	$-0.978090\pi$
$647$	− 89.0000i	− 0.137558i	0.997632	−	0.0687790i	$-0.0219103\pi$
$648$	0	0
$649$	−289.000	−0.445300
$650$	0	0
$651$	0	0
$652$	0	0
$653$	527.000	0.807044	0.403522	−	0.914970i	$-0.367786\pi$
$653$	527.000	0.807044	0.403522	+	0.914970i	$0.367786\pi$
$654$	0	0
$655$	− 185.000i	− 0.282443i
$656$	0	0
$657$	−760.000	−1.15677
$658$	0	0
$659$	936.000i	1.42033i	0.704033	+	0.710167i	$0.251381\pi$
$659$	936.000i	1.42033i	−0.704033	+	0.710167i	$0.748619\pi$
$660$	0	0
$661$	745.000	1.12708	0.563540	−	0.826089i	$-0.309439\pi$
$661$	745.000	1.12708	0.563540	+	0.826089i	$0.309439\pi$
$662$	0	0
$663$	− 24.0000i	− 0.0361991i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	168.000i	0.251874i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 17.0000i	− 0.0253353i
$672$	0	0
$673$	−720.000	−1.06984	−0.534918	−	0.844904i	$-0.679658\pi$
$673$	−720.000	−1.06984	−0.534918	+	0.844904i	$0.679658\pi$
$674$	0	0
$675$	408.000i	0.604444i
$676$	0	0
$677$	1151.00	1.70015	0.850074	−	0.526663i	$-0.176557\pi$
$677$	1151.00	1.70015	0.850074	+	0.526663i	$0.176557\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−415.000	−0.609398
$682$	0	0
$683$	− 737.000i	− 1.07906i	−0.841965	−	0.539531i	$-0.818601\pi$
$683$	− 737.000i	− 1.07906i	0.841965	−	0.539531i	$-0.181399\pi$
$684$	0	0
$685$	143.000	0.208759
$686$	0	0
$687$	− 143.000i	− 0.208151i
$688$	0	0
$689$	600.000	0.870827
$690$	0	0
$691$	− 1015.00i	− 1.46889i	−0.678671	−	0.734443i	$-0.737444\pi$
$691$	− 1015.00i	− 1.46889i	0.678671	−	0.734443i	$-0.262556\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	216.000i	0.310791i
$696$	0	0
$697$	−48.0000	−0.0688666
$698$	0	0
$699$	− 289.000i	− 0.413448i
$700$	0	0
$701$	1226.00	1.74893	0.874465	−	0.485089i	$-0.161213\pi$
$701$	1226.00	1.74893	0.874465	+	0.485089i	$0.161213\pi$
$702$	0	0
$703$	− 343.000i	− 0.487909i
$704$	0	0
$705$	−55.0000	−0.0780142
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−625.000	−0.881523	−0.440762	−	0.897624i	$-0.645292\pi$
$709$	−625.000	−0.881523	−0.440762	+	0.897624i	$0.645292\pi$
$710$	0	0
$711$	− 328.000i	− 0.461322i
$712$	0	0
$713$	−287.000	−0.402525
$714$	0	0
$715$	− 408.000i	− 0.570629i
$716$	0	0
$717$	−226.000	−0.315202
$718$	0	0
$719$	137.000i	0.190542i	0.995451	+	0.0952712i	$0.0303718\pi$
$719$	137.000i	0.190542i	−0.995451	+	0.0952712i	$0.969628\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	95.0000i	0.131397i
$724$	0	0
$725$	−576.000	−0.794483
$726$	0	0
$727$	− 960.000i	− 1.32050i	−0.751048	−	0.660248i	$-0.770451\pi$
$727$	− 960.000i	− 1.32050i	0.751048	−	0.660248i	$-0.229549\pi$
$728$	0	0
$729$	287.000	0.393690
$730$	0	0
$731$	24.0000i	0.0328317i
$732$	0	0
$733$	−479.000	−0.653479	−0.326739	−	0.945114i	$-0.605950\pi$
$733$	−479.000	−0.653479	−0.326739	+	0.945114i	$0.605950\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1105.00	1.49932
$738$	0	0
$739$	511.000i	0.691475i	0.938331	+	0.345737i	$0.112371\pi$
$739$	511.000i	0.691475i	−0.938331	+	0.345737i	$0.887629\pi$
$740$	0	0
$741$	−168.000	−0.226721
$742$	0	0
$743$	528.000i	0.710633i	0.934746	+	0.355316i	$0.115627\pi$
$743$	528.000i	0.710633i	−0.934746	+	0.355316i	$0.884373\pi$
$744$	0	0
$745$	47.0000	0.0630872
$746$	0	0
$747$	576.000i	0.771084i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	809.000i	1.07723i	0.842552	+	0.538615i	$0.181052\pi$
$751$	809.000i	1.07723i	−0.842552	+	0.538615i	$0.818948\pi$
$752$	0	0
$753$	−38.0000	−0.0504648
$754$	0	0
$755$	− 199.000i	− 0.263576i
$756$	0	0
$757$	−120.000	−0.158520	−0.0792602	−	0.996854i	$-0.525256\pi$
$757$	−120.000	−0.158520	−0.0792602	+	0.996854i	$0.525256\pi$
$758$	0	0
$759$	− 119.000i	− 0.156785i
$760$	0	0
$761$	−769.000	−1.01051	−0.505256	−	0.862969i	$-0.668602\pi$
$761$	−769.000	−1.01051	−0.505256	+	0.862969i	$0.668602\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000	0.0104575
$766$	0	0
$767$	408.000i	0.531943i
$768$	0	0
$769$	144.000	0.187256	0.0936281	−	0.995607i	$-0.470154\pi$
$769$	144.000	0.187256	0.0936281	+	0.995607i	$0.470154\pi$
$770$	0	0
$771$	193.000i	0.250324i
$772$	0	0
$773$	407.000	0.526520	0.263260	−	0.964725i	$-0.415202\pi$
$773$	407.000	0.526520	0.263260	+	0.964725i	$0.415202\pi$
$774$	0	0
$775$	− 984.000i	− 1.26968i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	336.000i	0.431322i
$780$	0	0
$781$	−1632.00	−2.08963
$782$	0	0
$783$	− 408.000i	− 0.521073i
$784$	0	0
$785$	−73.0000	−0.0929936
$786$	0	0
$787$	1135.00i	1.44219i	0.692839	+	0.721093i	$0.256360\pi$
$787$	1135.00i	1.44219i	−0.692839	+	0.721093i	$0.743640\pi$
$788$	0	0
$789$	233.000	0.295311
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.0000	−0.0302648
$794$	0	0
$795$	− 25.0000i	− 0.0314465i
$796$	0	0
$797$	−312.000	−0.391468	−0.195734	−	0.980657i	$-0.562709\pi$
$797$	−312.000	−0.391468	−0.195734	+	0.980657i	$0.562709\pi$
$798$	0	0
$799$	− 55.0000i	− 0.0688360i
$800$	0	0
$801$	−760.000	−0.948814
$802$	0	0
$803$	1615.00i	2.01121i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	455.000i	0.563817i
$808$	0	0
$809$	−1297.00	−1.60321	−0.801607	−	0.597851i	$-0.796021\pi$
$809$	−1297.00	−1.60321	−0.801607	+	0.597851i	$0.796021\pi$
$810$	0	0
$811$	− 1128.00i	− 1.39088i	−0.718586	−	0.695438i	$-0.755211\pi$
$811$	− 1128.00i	− 1.39088i	0.718586	−	0.695438i	$-0.244789\pi$
$812$	0	0
$813$	425.000	0.522755
$814$	0	0
$815$	− 55.0000i	− 0.0674847i
$816$	0	0
$817$	168.000	0.205630
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−383.000	−0.466504	−0.233252	−	0.972416i	$-0.574937\pi$
$821$	−383.000	−0.466504	−0.233252	+	0.972416i	$0.574937\pi$
$822$	0	0
$823$	− 55.0000i	− 0.0668287i	−0.999442	−	0.0334143i	$-0.989362\pi$
$823$	− 55.0000i	− 0.0668287i	0.999442	−	0.0334143i	$-0.0106381\pi$
$824$	0	0
$825$	408.000	0.494545
$826$	0	0
$827$	− 696.000i	− 0.841596i	−0.907154	−	0.420798i	$-0.861750\pi$
$827$	− 696.000i	− 0.841596i	0.907154	−	0.420798i	$-0.138250\pi$
$828$	0	0
$829$	−505.000	−0.609168	−0.304584	−	0.952486i	$-0.598517\pi$
$829$	−505.000	−0.609168	−0.304584	+	0.952486i	$0.598517\pi$
$830$	0	0
$831$	− 167.000i	− 0.200963i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 206.000i	− 0.246707i
$836$	0	0
$837$	697.000	0.832736
$838$	0	0
$839$	− 48.0000i	− 0.0572110i	−0.999591	−	0.0286055i	$-0.990893\pi$
$839$	− 48.0000i	− 0.0572110i	0.999591	−	0.0286055i	$-0.00910665\pi$
$840$	0	0
$841$	−265.000	−0.315101
$842$	0	0
$843$	432.000i	0.512456i
$844$	0	0
$845$	−407.000	−0.481657
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−223.000	−0.262662
$850$	0	0
$851$	− 343.000i	− 0.403055i
$852$	0	0
$853$	−696.000	−0.815944	−0.407972	−	0.912995i	$-0.633764\pi$
$853$	−696.000	−0.815944	−0.407972	+	0.912995i	$0.633764\pi$
$854$	0	0
$855$	− 56.0000i	− 0.0654971i
$856$	0	0
$857$	−1199.00	−1.39907	−0.699533	−	0.714600i	$-0.746609\pi$
$857$	−1199.00	−1.39907	−0.699533	+	0.714600i	$0.746609\pi$
$858$	0	0
$859$	775.000i	0.902212i	0.892470	+	0.451106i	$0.148970\pi$
$859$	775.000i	0.902212i	−0.892470	+	0.451106i	$0.851030\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	343.000i	0.397451i	0.980055	+	0.198725i	$0.0636802\pi$
$863$	343.000i	0.397451i	−0.980055	+	0.198725i	$0.936320\pi$
$864$	0	0
$865$	239.000	0.276301
$866$	0	0
$867$	288.000i	0.332180i
$868$	0	0
$869$	−697.000	−0.802071
$870$	0	0
$871$	− 1560.00i	− 1.79104i
$872$	0	0
$873$	−1152.00	−1.31959
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1151.00	−1.31243	−0.656214	−	0.754575i	$-0.727843\pi$
$877$	−1151.00	−1.31243	−0.656214	+	0.754575i	$0.727843\pi$
$878$	0	0
$879$	26.0000i	0.0295791i
$880$	0	0
$881$	866.000	0.982974	0.491487	−	0.870885i	$-0.336454\pi$
$881$	866.000	0.982974	0.491487	+	0.870885i	$0.336454\pi$
$882$	0	0
$883$	− 648.000i	− 0.733862i	−0.930248	−	0.366931i	$-0.880409\pi$
$883$	− 648.000i	− 0.733862i	0.930248	−	0.366931i	$-0.119591\pi$
$884$	0	0
$885$	17.0000	0.0192090
$886$	0	0
$887$	679.000i	0.765502i	0.923852	+	0.382751i	$0.125023\pi$
$887$	679.000i	0.765502i	−0.923852	+	0.382751i	$0.874977\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 935.000i	− 1.04938i
$892$	0	0
$893$	−385.000	−0.431131
$894$	0	0
$895$	17.0000i	0.0189944i
$896$	0	0
$897$	−168.000	−0.187291
$898$	0	0
$899$	984.000i	1.09455i
$900$	0	0
$901$	25.0000	0.0277469
$902$	0	0
$903$	0	0
$904$	0	0
$905$	70.0000	0.0773481
$906$	0	0
$907$	353.000i	0.389195i	0.980883	+	0.194598i	$0.0623401\pi$
$907$	353.000i	0.389195i	−0.980883	+	0.194598i	$0.937660\pi$
$908$	0	0
$909$	−584.000	−0.642464
$910$	0	0
$911$	− 144.000i	− 0.158068i	−0.996872	−	0.0790340i	$-0.974816\pi$
$911$	− 144.000i	− 0.158068i	0.996872	−	0.0790340i	$-0.0251836\pi$
$912$	0	0
$913$	1224.00	1.34064
$914$	0	0
$915$	1.00000i	0.00109290i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1097.00i	1.19369i	0.802357	+	0.596844i	$0.203579\pi$
$919$	1097.00i	1.19369i	−0.802357	+	0.596844i	$0.796421\pi$
$920$	0	0
$921$	−264.000	−0.286645
$922$	0	0
$923$	2304.00i	2.49621i
$924$	0	0
$925$	1176.00	1.27135
$926$	0	0
$927$	712.000i	0.768069i
$928$	0	0
$929$	529.000	0.569429	0.284715	−	0.958612i	$-0.408101\pi$
$929$	529.000	0.569429	0.284715	+	0.958612i	$0.408101\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	343.000	0.367631
$934$	0	0
$935$	− 17.0000i	− 0.0181818i
$936$	0	0
$937$	−146.000	−0.155816	−0.0779082	−	0.996961i	$-0.524824\pi$
$937$	−146.000	−0.155816	−0.0779082	+	0.996961i	$0.524824\pi$
$938$	0	0
$939$	335.000i	0.356763i
$940$	0	0
$941$	−743.000	−0.789586	−0.394793	−	0.918770i	$-0.629184\pi$
$941$	−743.000	−0.789586	−0.394793	+	0.918770i	$0.629184\pi$
$942$	0	0
$943$	336.000i	0.356310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 521.000i	− 0.550158i	−0.961422	−	0.275079i	$-0.911296\pi$
$947$	− 521.000i	− 0.550158i	0.961422	−	0.275079i	$-0.0887041\pi$
$948$	0	0
$949$	2280.00	2.40253
$950$	0	0
$951$	529.000i	0.556257i
$952$	0	0
$953$	1440.00	1.51102	0.755509	−	0.655138i	$-0.227390\pi$
$953$	1440.00	1.51102	0.755509	+	0.655138i	$0.227390\pi$
$954$	0	0
$955$	199.000i	0.208377i
$956$	0	0
$957$	−408.000	−0.426332
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−720.000	−0.749220
$962$	0	0
$963$	1480.00i	1.53686i
$964$	0	0
$965$	47.0000	0.0487047
$966$	0	0
$967$	− 1920.00i	− 1.98552i	−0.120106	−	0.992761i	$-0.538323\pi$
$967$	− 1920.00i	− 1.98552i	0.120106	−	0.992761i	$-0.461677\pi$
$968$	0	0
$969$	−7.00000	−0.00722394
$970$	0	0
$971$	329.000i	0.338826i	0.985545	+	0.169413i	$0.0541872\pi$
$971$	329.000i	0.338826i	−0.985545	+	0.169413i	$0.945813\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 576.000i	− 0.590769i
$976$	0	0
$977$	721.000	0.737973	0.368987	−	0.929435i	$-0.379705\pi$
$977$	721.000	0.737973	0.368987	+	0.929435i	$0.379705\pi$
$978$	0	0
$979$	1615.00i	1.64964i
$980$	0	0
$981$	−568.000	−0.579001
$982$	0	0
$983$	− 631.000i	− 0.641913i	−0.947094	−	0.320956i	$-0.895996\pi$
$983$	− 631.000i	− 0.641913i	0.947094	−	0.320956i	$-0.104004\pi$
$984$	0	0
$985$	−24.0000	−0.0243655
$986$	0	0
$987$	0	0
$988$	0	0
$989$	168.000	0.169869
$990$	0	0
$991$	− 727.000i	− 0.733602i	−0.930299	−	0.366801i	$-0.880453\pi$
$991$	− 727.000i	− 0.733602i	0.930299	−	0.366801i	$-0.119547\pi$
$992$	0	0
$993$	55.0000	0.0553877
$994$	0	0
$995$	− 137.000i	− 0.137688i
$996$	0	0
$997$	25.0000	0.0250752	0.0125376	−	0.999921i	$-0.496009\pi$
$997$	25.0000	0.0250752	0.0125376	+	0.999921i	$0.496009\pi$
$998$	0	0
$999$	833.000i	0.833834i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.d.c.1471.1		2
4.3	odd	2	inner	1568.3.d.c.1471.2		2
7.3	odd	6		224.3.r.b.191.2	yes	4
7.5	odd	6		224.3.r.b.95.1	✓	4
7.6	odd	2		1568.3.d.d.1471.2		2
28.3	even	6		224.3.r.b.191.1	yes	4
28.19	even	6		224.3.r.b.95.2	yes	4
28.27	even	2		1568.3.d.d.1471.1		2
56.3	even	6		448.3.r.b.191.2		4
56.5	odd	6		448.3.r.b.319.2		4
56.19	even	6		448.3.r.b.319.1		4
56.45	odd	6		448.3.r.b.191.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.r.b.95.1	✓	4	7.5	odd	6
224.3.r.b.95.2	yes	4	28.19	even	6
224.3.r.b.191.1	yes	4	28.3	even	6
224.3.r.b.191.2	yes	4	7.3	odd	6
448.3.r.b.191.1		4	56.45	odd	6
448.3.r.b.191.2		4	56.3	even	6
448.3.r.b.319.1		4	56.19	even	6
448.3.r.b.319.2		4	56.5	odd	6
1568.3.d.c.1471.1		2	1.1	even	1	trivial
1568.3.d.c.1471.2		2	4.3	odd	2	inner
1568.3.d.d.1471.1		2	28.27	even	2
1568.3.d.d.1471.2		2	7.6	odd	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000 q^{5} +8.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000 q^{5} +8.00000 q^{9} -17.0000i q^{11} -24.0000 q^{13} +1.00000i q^{15} -1.00000 q^{17} +7.00000i q^{19} +7.00000i q^{23} -24.0000 q^{25} -17.0000i q^{27} +24.0000 q^{29} +41.0000i q^{31} -17.0000 q^{33} -49.0000 q^{37} +24.0000i q^{39} +48.0000 q^{41} -24.0000i q^{43} -8.00000 q^{45} +55.0000i q^{47} +1.00000i q^{51} -25.0000 q^{53} +17.0000i q^{55} +7.00000 q^{57} -17.0000i q^{59} +1.00000 q^{61} +24.0000 q^{65} +65.0000i q^{67} +7.00000 q^{69} -96.0000i q^{71} -95.0000 q^{73} +24.0000i q^{75} -41.0000i q^{79} +55.0000 q^{81} +72.0000i q^{83} +1.00000 q^{85} -24.0000i q^{87} -95.0000 q^{89} +41.0000 q^{93} -7.00000i q^{95} -144.000 q^{97} -136.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 16 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 16 * q^9	\( 2 q - 2 q^{5} + 16 q^{9} - 48 q^{13} - 2 q^{17} - 48 q^{25} + 48 q^{29} - 34 q^{33} - 98 q^{37} + 96 q^{41} - 16 q^{45} - 50 q^{53} + 14 q^{57} + 2 q^{61} + 48 q^{65} + 14 q^{69} - 190 q^{73} + 110 q^{81} + 2 q^{85} - 190 q^{89} + 82 q^{93} - 288 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 16 * q^9 - 48 * q^13 - 2 * q^17 - 48 * q^25 + 48 * q^29 - 34 * q^33 - 98 * q^37 + 96 * q^41 - 16 * q^45 - 50 * q^53 + 14 * q^57 + 2 * q^61 + 48 * q^65 + 14 * q^69 - 190 * q^73 + 110 * q^81 + 2 * q^85 - 190 * q^89 + 82 * q^93 - 288 * q^97

Introduction

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