LMFDB - Embedded newform 1568.3.g.l.687.3

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:	$6$
Coefficient field:	6.0.15582448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 $ x^6 - 3x^5 + 13x^4 - 21x^3 + 20x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			687.3
Root			$0.500000 - 2.94141i$ of defining polynomial
Character	$\chi$	$=$	1568.687
Dual form			1568.3.g.l.687.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.64878 q^{3} -4.56111i q^{5} -6.28154 q^{9} +O(q^{10})$	$q+1.64878 q^{3} -4.56111i q^{5} -6.28154 q^{9} +12.3797 q^{11} +18.3741i q^{13} -7.52026i q^{15} +13.0284 q^{17} -3.02524 q^{19} +30.3237i q^{23} +4.19624 q^{25} -25.1958 q^{27} +22.7701i q^{29} -22.5608i q^{31} +20.4113 q^{33} +13.7797i q^{37} +30.2948i q^{39} +60.5026 q^{41} -39.0188 q^{43} +28.6508i q^{45} -20.3360i q^{47} +21.4810 q^{51} -4.76596i q^{53} -56.4650i q^{55} -4.98794 q^{57} +11.7377 q^{59} +108.904i q^{61} +83.8066 q^{65} +79.1994 q^{67} +49.9970i q^{69} +12.9952i q^{71} +98.5819 q^{73} +6.91866 q^{75} -131.225i q^{79} +14.9915 q^{81} +28.3732 q^{83} -59.4242i q^{85} +37.5428i q^{87} +157.418 q^{89} -37.1977i q^{93} +13.7985i q^{95} +39.6175 q^{97} -77.7633 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 40 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 40 * q^9	$ 6 q + 6 q^{3} + 40 q^{9} + 30 q^{11} + 30 q^{17} - 78 q^{19} + 92 q^{25} + 78 q^{27} - 78 q^{33} + 116 q^{41} + 100 q^{43} + 10 q^{51} + 166 q^{57} + 110 q^{59} + 32 q^{65} + 434 q^{67} + 102 q^{73} + 60 q^{75} + 82 q^{81} - 268 q^{83} + 214 q^{89} + 76 q^{97} - 252 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 40 * q^9 + 30 * q^11 + 30 * q^17 - 78 * q^19 + 92 * q^25 + 78 * q^27 - 78 * q^33 + 116 * q^41 + 100 * q^43 + 10 * q^51 + 166 * q^57 + 110 * q^59 + 32 * q^65 + 434 * q^67 + 102 * q^73 + 60 * q^75 + 82 * q^81 - 268 * q^83 + 214 * q^89 + 76 * q^97 - 252 * 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
$3$	1.64878	0.549592	0.274796	−	0.961503i	$-0.411390\pi$
$3$	1.64878	0.549592	0.274796	+	0.961503i	$0.411390\pi$
$4$	0	0
$5$	− 4.56111i	− 0.912223i	−0.889923	−	0.456111i	$-0.849242\pi$
$5$	− 4.56111i	− 0.912223i	0.889923	−	0.456111i	$-0.150758\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−6.28154	−0.697949
$10$	0	0
$11$	12.3797	1.12542	0.562712	−	0.826653i	$-0.309758\pi$
$11$	12.3797	1.12542	0.562712	+	0.826653i	$0.309758\pi$
$12$	0	0
$13$	18.3741i	1.41340i	0.707516	+	0.706698i	$0.249816\pi$
$13$	18.3741i	1.41340i	−0.707516	+	0.706698i	$0.750184\pi$
$14$	0	0
$15$	− 7.52026i	− 0.501350i
$16$	0	0
$17$	13.0284	0.766378	0.383189	−	0.923670i	$-0.374826\pi$
$17$	13.0284	0.766378	0.383189	+	0.923670i	$0.374826\pi$
$18$	0	0
$19$	−3.02524	−0.159223	−0.0796115	−	0.996826i	$-0.525368\pi$
$19$	−3.02524	−0.159223	−0.0796115	+	0.996826i	$0.525368\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	30.3237i	1.31842i	0.751958	+	0.659211i	$0.229110\pi$
$23$	30.3237i	1.31842i	−0.751958	+	0.659211i	$0.770890\pi$
$24$	0	0
$25$	4.19624	0.167850
$26$	0	0
$27$	−25.1958	−0.933179
$28$	0	0
$29$	22.7701i	0.785176i	0.919714	+	0.392588i	$0.128420\pi$
$29$	22.7701i	0.785176i	−0.919714	+	0.392588i	$0.871580\pi$
$30$	0	0
$31$	− 22.5608i	− 0.727767i	−0.931445	−	0.363883i	$-0.881451\pi$
$31$	− 22.5608i	− 0.727767i	0.931445	−	0.363883i	$-0.118549\pi$
$32$	0	0
$33$	20.4113	0.618524
$34$	0	0
$35$	0	0
$36$	0	0
$37$	13.7797i	0.372423i	0.982510	+	0.186212i	$0.0596210\pi$
$37$	13.7797i	0.372423i	−0.982510	+	0.186212i	$0.940379\pi$
$38$	0	0
$39$	30.2948i	0.776791i
$40$	0	0
$41$	60.5026	1.47567	0.737837	−	0.674979i	$-0.235847\pi$
$41$	60.5026	1.47567	0.737837	+	0.674979i	$0.235847\pi$
$42$	0	0
$43$	−39.0188	−0.907414	−0.453707	−	0.891151i	$-0.649899\pi$
$43$	−39.0188	−0.907414	−0.453707	+	0.891151i	$0.649899\pi$
$44$	0	0
$45$	28.6508i	0.636685i
$46$	0	0
$47$	− 20.3360i	− 0.432681i	−0.976318	−	0.216341i	$-0.930588\pi$
$47$	− 20.3360i	− 0.432681i	0.976318	−	0.216341i	$-0.0694122\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	21.4810	0.421195
$52$	0	0
$53$	− 4.76596i	− 0.0899237i	−0.998989	−	0.0449619i	$-0.985683\pi$
$53$	− 4.76596i	− 0.0899237i	0.998989	−	0.0449619i	$-0.0143166\pi$
$54$	0	0
$55$	− 56.4650i	− 1.02664i
$56$	0	0
$57$	−4.98794	−0.0875077
$58$	0	0
$59$	11.7377	0.198944	0.0994718	−	0.995040i	$-0.468285\pi$
$59$	11.7377	0.198944	0.0994718	+	0.995040i	$0.468285\pi$
$60$	0	0
$61$	108.904i	1.78531i	0.450738	+	0.892656i	$0.351161\pi$
$61$	108.904i	1.78531i	−0.450738	+	0.892656i	$0.648839\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	83.8066	1.28933
$66$	0	0
$67$	79.1994	1.18208	0.591041	−	0.806642i	$-0.298717\pi$
$67$	79.1994	1.18208	0.591041	+	0.806642i	$0.298717\pi$
$68$	0	0
$69$	49.9970i	0.724595i
$70$	0	0
$71$	12.9952i	0.183031i	0.995804	+	0.0915157i	$0.0291712\pi$
$71$	12.9952i	0.183031i	−0.995804	+	0.0915157i	$0.970829\pi$
$72$	0	0
$73$	98.5819	1.35044	0.675218	−	0.737618i	$-0.264049\pi$
$73$	98.5819	1.35044	0.675218	+	0.737618i	$0.264049\pi$
$74$	0	0
$75$	6.91866	0.0922488
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 131.225i	− 1.66107i	−0.556966	−	0.830535i	$-0.688035\pi$
$79$	− 131.225i	− 1.66107i	0.556966	−	0.830535i	$-0.311965\pi$
$80$	0	0
$81$	14.9915	0.185081
$82$	0	0
$83$	28.3732	0.341846	0.170923	−	0.985284i	$-0.445325\pi$
$83$	28.3732	0.341846	0.170923	+	0.985284i	$0.445325\pi$
$84$	0	0
$85$	− 59.4242i	− 0.699108i
$86$	0	0
$87$	37.5428i	0.431526i
$88$	0	0
$89$	157.418	1.76874	0.884371	−	0.466785i	$-0.154588\pi$
$89$	157.418	1.76874	0.884371	+	0.466785i	$0.154588\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 37.1977i	− 0.399975i
$94$	0	0
$95$	13.7985i	0.145247i
$96$	0	0
$97$	39.6175	0.408428	0.204214	−	0.978926i	$-0.434536\pi$
$97$	39.6175	0.408428	0.204214	+	0.978926i	$0.434536\pi$
$98$	0	0
$99$	−77.7633	−0.785488
$100$	0	0
$101$	43.6183i	0.431864i	0.976408	+	0.215932i	$0.0692790\pi$
$101$	43.6183i	0.431864i	−0.976408	+	0.215932i	$0.930721\pi$
$102$	0	0
$103$	62.9020i	0.610699i	0.952240	+	0.305350i	$0.0987734\pi$
$103$	62.9020i	0.610699i	−0.952240	+	0.305350i	$0.901227\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−44.2267	−0.413334	−0.206667	−	0.978411i	$-0.566262\pi$
$107$	−44.2267	−0.413334	−0.206667	+	0.978411i	$0.566262\pi$
$108$	0	0
$109$	− 8.81520i	− 0.0808734i	−0.999182	−	0.0404367i	$-0.987125\pi$
$109$	− 8.81520i	− 0.0808734i	0.999182	−	0.0404367i	$-0.0128749\pi$
$110$	0	0
$111$	22.7196i	0.204681i
$112$	0	0
$113$	−121.408	−1.07440	−0.537202	−	0.843454i	$-0.680519\pi$
$113$	−121.408	−1.07440	−0.537202	+	0.843454i	$0.680519\pi$
$114$	0	0
$115$	138.310	1.20269
$116$	0	0
$117$	− 115.418i	− 0.986477i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	32.2559	0.266578
$122$	0	0
$123$	99.7553	0.811018
$124$	0	0
$125$	− 133.167i	− 1.06534i
$126$	0	0
$127$	− 222.845i	− 1.75468i	−0.479868	−	0.877341i	$-0.659315\pi$
$127$	− 222.845i	− 1.75468i	0.479868	−	0.877341i	$-0.340685\pi$
$128$	0	0
$129$	−64.3333	−0.498708
$130$	0	0
$131$	237.054	1.80957	0.904785	−	0.425869i	$-0.140032\pi$
$131$	237.054	1.80957	0.904785	+	0.425869i	$0.140032\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	114.921i	0.851267i
$136$	0	0
$137$	−9.66276	−0.0705311	−0.0352656	−	0.999378i	$-0.511228\pi$
$137$	−9.66276	−0.0705311	−0.0352656	+	0.999378i	$0.511228\pi$
$138$	0	0
$139$	−63.0621	−0.453684	−0.226842	−	0.973932i	$-0.572840\pi$
$139$	−63.0621	−0.453684	−0.226842	+	0.973932i	$0.572840\pi$
$140$	0	0
$141$	− 33.5296i	− 0.237798i
$142$	0	0
$143$	227.466i	1.59067i
$144$	0	0
$145$	103.857	0.716255
$146$	0	0
$147$	0	0
$148$	0	0
$149$	269.912i	1.81149i	0.423820	+	0.905746i	$0.360689\pi$
$149$	269.912i	1.81149i	−0.423820	+	0.905746i	$0.639311\pi$
$150$	0	0
$151$	108.178i	0.716408i	0.933643	+	0.358204i	$0.116611\pi$
$151$	108.178i	0.716408i	−0.933643	+	0.358204i	$0.883389\pi$
$152$	0	0
$153$	−81.8386	−0.534893
$154$	0	0
$155$	−102.902	−0.663885
$156$	0	0
$157$	− 118.432i	− 0.754343i	−0.926144	−	0.377171i	$-0.876897\pi$
$157$	− 118.432i	− 0.754343i	0.926144	−	0.377171i	$-0.123103\pi$
$158$	0	0
$159$	− 7.85800i	− 0.0494214i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	82.0284	0.503242	0.251621	−	0.967826i	$-0.419036\pi$
$163$	82.0284	0.503242	0.251621	+	0.967826i	$0.419036\pi$
$164$	0	0
$165$	− 93.0982i	− 0.564231i
$166$	0	0
$167$	131.596i	0.788002i	0.919110	+	0.394001i	$0.128909\pi$
$167$	131.596i	0.788002i	−0.919110	+	0.394001i	$0.871091\pi$
$168$	0	0
$169$	−168.609	−0.997687
$170$	0	0
$171$	19.0031	0.111130
$172$	0	0
$173$	110.114i	0.636494i	0.948008	+	0.318247i	$0.103094\pi$
$173$	110.114i	0.636494i	−0.948008	+	0.318247i	$0.896906\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	19.3528	0.109338
$178$	0	0
$179$	−153.853	−0.859512	−0.429756	−	0.902945i	$-0.641400\pi$
$179$	−153.853	−0.859512	−0.429756	+	0.902945i	$0.641400\pi$
$180$	0	0
$181$	227.511i	1.25697i	0.777823	+	0.628484i	$0.216324\pi$
$181$	227.511i	1.25697i	−0.777823	+	0.628484i	$0.783676\pi$
$182$	0	0
$183$	179.558i	0.981194i
$184$	0	0
$185$	62.8506	0.339733
$186$	0	0
$187$	161.288	0.862500
$188$	0	0
$189$	0	0
$190$	0	0
$191$	121.546i	0.636364i	0.948030	+	0.318182i	$0.103072\pi$
$191$	121.546i	0.636364i	−0.948030	+	0.318182i	$0.896928\pi$
$192$	0	0
$193$	85.4552	0.442773	0.221386	−	0.975186i	$-0.428942\pi$
$193$	85.4552	0.442773	0.221386	+	0.975186i	$0.428942\pi$
$194$	0	0
$195$	138.178	0.708606
$196$	0	0
$197$	− 214.100i	− 1.08680i	−0.839474	−	0.543400i	$-0.817137\pi$
$197$	− 214.100i	− 1.08680i	0.839474	−	0.543400i	$-0.182863\pi$
$198$	0	0
$199$	− 248.223i	− 1.24735i	−0.781682	−	0.623677i	$-0.785638\pi$
$199$	− 248.223i	− 1.24735i	0.781682	−	0.623677i	$-0.214362\pi$
$200$	0	0
$201$	130.582	0.649662
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 275.959i	− 1.34614i
$206$	0	0
$207$	− 190.480i	− 0.920191i
$208$	0	0
$209$	−37.4514	−0.179193
$210$	0	0
$211$	−191.753	−0.908783	−0.454392	−	0.890802i	$-0.650143\pi$
$211$	−191.753	−0.908783	−0.454392	+	0.890802i	$0.650143\pi$
$212$	0	0
$213$	21.4262i	0.100593i
$214$	0	0
$215$	177.969i	0.827764i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	162.539	0.742189
$220$	0	0
$221$	239.386i	1.08320i
$222$	0	0
$223$	− 41.2269i	− 0.184874i	−0.995719	−	0.0924370i	$-0.970534\pi$
$223$	− 41.2269i	− 0.184874i	0.995719	−	0.0924370i	$-0.0294657\pi$
$224$	0	0
$225$	−26.3588	−0.117150
$226$	0	0
$227$	70.4437	0.310325	0.155162	−	0.987889i	$-0.450410\pi$
$227$	70.4437	0.310325	0.155162	+	0.987889i	$0.450410\pi$
$228$	0	0
$229$	94.5190i	0.412747i	0.978473	+	0.206373i	$0.0661661\pi$
$229$	94.5190i	0.412747i	−0.978473	+	0.206373i	$0.933834\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−136.287	−0.584922	−0.292461	−	0.956277i	$-0.594474\pi$
$233$	−136.287	−0.584922	−0.292461	+	0.956277i	$0.594474\pi$
$234$	0	0
$235$	−92.7550	−0.394702
$236$	0	0
$237$	− 216.360i	− 0.912911i
$238$	0	0
$239$	173.230i	0.724813i	0.932020	+	0.362406i	$0.118045\pi$
$239$	173.230i	0.724813i	−0.932020	+	0.362406i	$0.881955\pi$
$240$	0	0
$241$	328.923	1.36483	0.682413	−	0.730967i	$-0.260931\pi$
$241$	328.923	1.36483	0.682413	+	0.730967i	$0.260931\pi$
$242$	0	0
$243$	251.480	1.03490
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 55.5862i	− 0.225045i
$248$	0	0
$249$	46.7811	0.187876
$250$	0	0
$251$	160.255	0.638466	0.319233	−	0.947676i	$-0.396575\pi$
$251$	160.255	0.638466	0.319233	+	0.947676i	$0.396575\pi$
$252$	0	0
$253$	375.397i	1.48378i
$254$	0	0
$255$	− 97.9772i	− 0.384224i
$256$	0	0
$257$	−145.442	−0.565921	−0.282960	−	0.959132i	$-0.591316\pi$
$257$	−145.442	−0.565921	−0.282960	+	0.959132i	$0.591316\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 143.031i	− 0.548012i
$262$	0	0
$263$	− 202.785i	− 0.771045i	−0.922698	−	0.385523i	$-0.874021\pi$
$263$	− 202.785i	− 0.771045i	0.922698	−	0.385523i	$-0.125979\pi$
$264$	0	0
$265$	−21.7381	−0.0820305
$266$	0	0
$267$	259.547	0.972087
$268$	0	0
$269$	221.312i	0.822720i	0.911473	+	0.411360i	$0.134946\pi$
$269$	221.312i	0.822720i	−0.911473	+	0.411360i	$0.865054\pi$
$270$	0	0
$271$	117.376i	0.433122i	0.976269	+	0.216561i	$0.0694840\pi$
$271$	117.376i	0.433122i	−0.976269	+	0.216561i	$0.930516\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	51.9480	0.188902
$276$	0	0
$277$	255.509i	0.922416i	0.887292	+	0.461208i	$0.152584\pi$
$277$	255.509i	0.922416i	−0.887292	+	0.461208i	$0.847416\pi$
$278$	0	0
$279$	141.716i	0.507944i
$280$	0	0
$281$	−278.004	−0.989336	−0.494668	−	0.869082i	$-0.664710\pi$
$281$	−278.004	−0.989336	−0.494668	+	0.869082i	$0.664710\pi$
$282$	0	0
$283$	56.6896	0.200317	0.100158	−	0.994972i	$-0.468065\pi$
$283$	56.6896	0.200317	0.100158	+	0.994972i	$0.468065\pi$
$284$	0	0
$285$	22.7506i	0.0798266i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−119.260	−0.412664
$290$	0	0
$291$	65.3204	0.224469
$292$	0	0
$293$	− 287.871i	− 0.982493i	−0.871020	−	0.491247i	$-0.836541\pi$
$293$	− 287.871i	− 0.982493i	0.871020	−	0.491247i	$-0.163459\pi$
$294$	0	0
$295$	− 53.5369i	− 0.181481i
$296$	0	0
$297$	−311.916	−1.05022
$298$	0	0
$299$	−557.172	−1.86345
$300$	0	0
$301$	0	0
$302$	0	0
$303$	71.9168i	0.237349i
$304$	0	0
$305$	496.724	1.62860
$306$	0	0
$307$	53.6483	0.174750	0.0873750	−	0.996175i	$-0.472152\pi$
$307$	53.6483	0.174750	0.0873750	+	0.996175i	$0.472152\pi$
$308$	0	0
$309$	103.711i	0.335636i
$310$	0	0
$311$	− 105.108i	− 0.337968i	−0.985619	−	0.168984i	$-0.945951\pi$
$311$	− 105.108i	− 0.337968i	0.985619	−	0.168984i	$-0.0540487\pi$
$312$	0	0
$313$	−210.490	−0.672492	−0.336246	−	0.941774i	$-0.609157\pi$
$313$	−210.490	−0.672492	−0.336246	+	0.941774i	$0.609157\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	62.8880i	0.198385i	0.995068	+	0.0991925i	$0.0316259\pi$
$317$	62.8880i	0.198385i	−0.995068	+	0.0991925i	$0.968374\pi$
$318$	0	0
$319$	281.886i	0.883655i
$320$	0	0
$321$	−72.9199	−0.227165
$322$	0	0
$323$	−39.4141	−0.122025
$324$	0	0
$325$	77.1023i	0.237238i
$326$	0	0
$327$	− 14.5343i	− 0.0444474i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−196.579	−0.593893	−0.296947	−	0.954894i	$-0.595968\pi$
$331$	−196.579	−0.593893	−0.296947	+	0.954894i	$0.595968\pi$
$332$	0	0
$333$	− 86.5575i	− 0.259932i
$334$	0	0
$335$	− 361.238i	− 1.07832i
$336$	0	0
$337$	591.516	1.75524	0.877620	−	0.479358i	$-0.159130\pi$
$337$	591.516	1.75524	0.877620	+	0.479358i	$0.159130\pi$
$338$	0	0
$339$	−200.174	−0.590484
$340$	0	0
$341$	− 279.295i	− 0.819046i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	228.042	0.660992
$346$	0	0
$347$	−247.540	−0.713371	−0.356685	−	0.934225i	$-0.616093\pi$
$347$	−247.540	−0.713371	−0.356685	+	0.934225i	$0.616093\pi$
$348$	0	0
$349$	288.749i	0.827362i	0.910422	+	0.413681i	$0.135757\pi$
$349$	288.749i	0.827362i	−0.910422	+	0.413681i	$0.864243\pi$
$350$	0	0
$351$	− 462.952i	− 1.31895i
$352$	0	0
$353$	1.26966	0.00359677	0.00179839	−	0.999998i	$-0.499428\pi$
$353$	1.26966	0.00359677	0.00179839	+	0.999998i	$0.499428\pi$
$354$	0	0
$355$	59.2727	0.166965
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.4021i	0.0484738i	0.999706	+	0.0242369i	$0.00771560\pi$
$359$	17.4021i	0.0484738i	−0.999706	+	0.0242369i	$0.992284\pi$
$360$	0	0
$361$	−351.848	−0.974648
$362$	0	0
$363$	53.1828	0.146509
$364$	0	0
$365$	− 449.643i	− 1.23190i
$366$	0	0
$367$	530.032i	1.44423i	0.691773	+	0.722115i	$0.256830\pi$
$367$	530.032i	1.44423i	−0.691773	+	0.722115i	$0.743170\pi$
$368$	0	0
$369$	−380.049	−1.02994
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 118.613i	− 0.317998i	−0.987279	−	0.158999i	$-0.949173\pi$
$373$	− 118.613i	− 0.317998i	0.987279	−	0.158999i	$-0.0508267\pi$
$374$	0	0
$375$	− 219.563i	− 0.585502i
$376$	0	0
$377$	−418.381	−1.10976
$378$	0	0
$379$	345.947	0.912790	0.456395	−	0.889777i	$-0.349140\pi$
$379$	345.947	0.912790	0.456395	+	0.889777i	$0.349140\pi$
$380$	0	0
$381$	− 367.421i	− 0.964359i
$382$	0	0
$383$	− 404.873i	− 1.05711i	−0.848899	−	0.528555i	$-0.822734\pi$
$383$	− 404.873i	− 1.05711i	0.848899	−	0.528555i	$-0.177266\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	245.098	0.633328
$388$	0	0
$389$	− 134.397i	− 0.345493i	−0.984966	−	0.172747i	$-0.944736\pi$
$389$	− 134.397i	− 0.345493i	0.984966	−	0.172747i	$-0.0552642\pi$
$390$	0	0
$391$	395.070i	1.01041i
$392$	0	0
$393$	390.848	0.994525
$394$	0	0
$395$	−598.530	−1.51527
$396$	0	0
$397$	− 612.481i	− 1.54277i	−0.636367	−	0.771386i	$-0.719564\pi$
$397$	− 612.481i	− 1.54277i	0.636367	−	0.771386i	$-0.280436\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−126.247	−0.314830	−0.157415	−	0.987533i	$-0.550316\pi$
$401$	−126.247	−0.314830	−0.157415	+	0.987533i	$0.550316\pi$
$402$	0	0
$403$	414.535	1.02862
$404$	0	0
$405$	− 68.3781i	− 0.168835i
$406$	0	0
$407$	170.588i	0.419134i
$408$	0	0
$409$	−342.519	−0.837455	−0.418727	−	0.908112i	$-0.637524\pi$
$409$	−342.519	−0.837455	−0.418727	+	0.908112i	$0.637524\pi$
$410$	0	0
$411$	−15.9317	−0.0387634
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 129.414i	− 0.311840i
$416$	0	0
$417$	−103.975	−0.249341
$418$	0	0
$419$	−376.392	−0.898311	−0.449155	−	0.893454i	$-0.648275\pi$
$419$	−376.392	−0.898311	−0.449155	+	0.893454i	$0.648275\pi$
$420$	0	0
$421$	− 111.135i	− 0.263978i	−0.991251	−	0.131989i	$-0.957864\pi$
$421$	− 111.135i	− 0.263978i	0.991251	−	0.131989i	$-0.0421363\pi$
$422$	0	0
$423$	127.742i	0.301989i
$424$	0	0
$425$	54.6704	0.128636
$426$	0	0
$427$	0	0
$428$	0	0
$429$	375.040i	0.874219i
$430$	0	0
$431$	− 41.2784i	− 0.0957734i	−0.998853	−	0.0478867i	$-0.984751\pi$
$431$	− 41.2784i	− 0.0957734i	0.998853	−	0.0478867i	$-0.0152486\pi$
$432$	0	0
$433$	675.176	1.55930	0.779649	−	0.626217i	$-0.215397\pi$
$433$	675.176	1.55930	0.779649	+	0.626217i	$0.215397\pi$
$434$	0	0
$435$	171.237	0.393648
$436$	0	0
$437$	− 91.7365i	− 0.209923i
$438$	0	0
$439$	− 530.256i	− 1.20787i	−0.797032	−	0.603936i	$-0.793598\pi$
$439$	− 530.256i	− 1.20787i	0.797032	−	0.603936i	$-0.206402\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−332.033	−0.749510	−0.374755	−	0.927124i	$-0.622273\pi$
$443$	−332.033	−0.749510	−0.374755	+	0.927124i	$0.622273\pi$
$444$	0	0
$445$	− 718.002i	− 1.61349i
$446$	0	0
$447$	445.025i	0.995582i
$448$	0	0
$449$	−19.4200	−0.0432517	−0.0216259	−	0.999766i	$-0.506884\pi$
$449$	−19.4200	−0.0432517	−0.0216259	+	0.999766i	$0.506884\pi$
$450$	0	0
$451$	749.002	1.66076
$452$	0	0
$453$	178.361i	0.393732i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−177.959	−0.389408	−0.194704	−	0.980862i	$-0.562375\pi$
$457$	−177.959	−0.389408	−0.194704	+	0.980862i	$0.562375\pi$
$458$	0	0
$459$	−328.262	−0.715168
$460$	0	0
$461$	299.341i	0.649329i	0.945829	+	0.324664i	$0.105251\pi$
$461$	299.341i	0.649329i	−0.945829	+	0.324664i	$0.894749\pi$
$462$	0	0
$463$	505.213i	1.09117i	0.838055	+	0.545586i	$0.183693\pi$
$463$	505.213i	1.09117i	−0.838055	+	0.545586i	$0.816307\pi$
$464$	0	0
$465$	−169.663	−0.364866
$466$	0	0
$467$	−650.324	−1.39256	−0.696278	−	0.717772i	$-0.745162\pi$
$467$	−650.324	−1.39256	−0.696278	+	0.717772i	$0.745162\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 195.268i	− 0.414581i
$472$	0	0
$473$	−483.039	−1.02122
$474$	0	0
$475$	−12.6946	−0.0267255
$476$	0	0
$477$	29.9375i	0.0627621i
$478$	0	0
$479$	− 660.951i	− 1.37986i	−0.723878	−	0.689928i	$-0.757642\pi$
$479$	− 660.951i	− 1.37986i	0.723878	−	0.689928i	$-0.242358\pi$
$480$	0	0
$481$	−253.190	−0.526382
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 180.700i	− 0.372577i
$486$	0	0
$487$	386.100i	0.792813i	0.918075	+	0.396407i	$0.129743\pi$
$487$	386.100i	0.792813i	−0.918075	+	0.396407i	$0.870257\pi$
$488$	0	0
$489$	135.246	0.276578
$490$	0	0
$491$	898.359	1.82965	0.914826	−	0.403848i	$-0.132328\pi$
$491$	898.359	1.82965	0.914826	+	0.403848i	$0.132328\pi$
$492$	0	0
$493$	296.659i	0.601742i
$494$	0	0
$495$	354.687i	0.716540i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−791.176	−1.58552	−0.792761	−	0.609532i	$-0.791357\pi$
$499$	−791.176	−1.58552	−0.792761	+	0.609532i	$0.791357\pi$
$500$	0	0
$501$	216.973i	0.433080i
$502$	0	0
$503$	− 798.990i	− 1.58845i	−0.607624	−	0.794225i	$-0.707877\pi$
$503$	− 798.990i	− 1.58845i	0.607624	−	0.794225i	$-0.292123\pi$
$504$	0	0
$505$	198.948	0.393956
$506$	0	0
$507$	−277.999	−0.548321
$508$	0	0
$509$	551.106i	1.08272i	0.840790	+	0.541362i	$0.182091\pi$
$509$	551.106i	1.08272i	−0.840790	+	0.541362i	$0.817909\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	76.2234	0.148584
$514$	0	0
$515$	286.903	0.557094
$516$	0	0
$517$	− 251.753i	− 0.486950i
$518$	0	0
$519$	181.553i	0.349812i
$520$	0	0
$521$	−142.783	−0.274055	−0.137028	−	0.990567i	$-0.543755\pi$
$521$	−142.783	−0.274055	−0.137028	+	0.990567i	$0.543755\pi$
$522$	0	0
$523$	−832.511	−1.59180	−0.795900	−	0.605429i	$-0.793002\pi$
$523$	−832.511	−1.59180	−0.795900	+	0.605429i	$0.793002\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 293.931i	− 0.557745i
$528$	0	0
$529$	−390.528	−0.738238
$530$	0	0
$531$	−73.7306	−0.138852
$532$	0	0
$533$	1111.68i	2.08571i
$534$	0	0
$535$	201.723i	0.377052i
$536$	0	0
$537$	−253.669	−0.472381
$538$	0	0
$539$	0	0
$540$	0	0
$541$	616.465i	1.13949i	0.821821	+	0.569746i	$0.192958\pi$
$541$	616.465i	1.13949i	−0.821821	+	0.569746i	$0.807042\pi$
$542$	0	0
$543$	375.115i	0.690819i
$544$	0	0
$545$	−40.2071	−0.0737746
$546$	0	0
$547$	−577.704	−1.05613	−0.528065	−	0.849204i	$-0.677082\pi$
$547$	−577.704	−1.05613	−0.528065	+	0.849204i	$0.677082\pi$
$548$	0	0
$549$	− 684.085i	− 1.24606i
$550$	0	0
$551$	− 68.8850i	− 0.125018i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	103.627	0.186715
$556$	0	0
$557$	514.710i	0.924075i	0.886860	+	0.462038i	$0.152881\pi$
$557$	514.710i	0.924075i	−0.886860	+	0.462038i	$0.847119\pi$
$558$	0	0
$559$	− 716.937i	− 1.28253i
$560$	0	0
$561$	265.927	0.474023
$562$	0	0
$563$	−608.719	−1.08121	−0.540603	−	0.841278i	$-0.681804\pi$
$563$	−608.719	−1.08121	−0.540603	+	0.841278i	$0.681804\pi$
$564$	0	0
$565$	553.754i	0.980096i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	186.374	0.327547	0.163774	−	0.986498i	$-0.447633\pi$
$569$	186.374	0.327547	0.163774	+	0.986498i	$0.447633\pi$
$570$	0	0
$571$	−183.776	−0.321849	−0.160924	−	0.986967i	$-0.551448\pi$
$571$	−183.776	−0.321849	−0.160924	+	0.986967i	$0.551448\pi$
$572$	0	0
$573$	200.401i	0.349741i
$574$	0	0
$575$	127.246i	0.221297i
$576$	0	0
$577$	−134.456	−0.233026	−0.116513	−	0.993189i	$-0.537172\pi$
$577$	−134.456	−0.233026	−0.116513	+	0.993189i	$0.537172\pi$
$578$	0	0
$579$	140.896	0.243345
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 59.0009i	− 0.101202i
$584$	0	0
$585$	−526.434	−0.899887
$586$	0	0
$587$	−921.405	−1.56968	−0.784842	−	0.619696i	$-0.787256\pi$
$587$	−921.405	−1.56968	−0.784842	+	0.619696i	$0.787256\pi$
$588$	0	0
$589$	68.2517i	0.115877i
$590$	0	0
$591$	− 353.002i	− 0.597297i
$592$	0	0
$593$	−96.3746	−0.162520	−0.0812602	−	0.996693i	$-0.525894\pi$
$593$	−96.3746	−0.162520	−0.0812602	+	0.996693i	$0.525894\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 409.265i	− 0.685536i
$598$	0	0
$599$	− 167.098i	− 0.278961i	−0.990225	−	0.139481i	$-0.955457\pi$
$599$	− 167.098i	− 0.278961i	0.990225	−	0.139481i	$-0.0445433\pi$
$600$	0	0
$601$	−88.4635	−0.147194	−0.0735969	−	0.997288i	$-0.523448\pi$
$601$	−88.4635	−0.147194	−0.0735969	+	0.997288i	$0.523448\pi$
$602$	0	0
$603$	−497.494	−0.825032
$604$	0	0
$605$	− 147.123i	− 0.243178i
$606$	0	0
$607$	55.5692i	0.0915472i	0.998952	+	0.0457736i	$0.0145753\pi$
$607$	55.5692i	0.0915472i	−0.998952	+	0.0457736i	$0.985425\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	373.657	0.611550
$612$	0	0
$613$	− 869.117i	− 1.41781i	−0.705304	−	0.708905i	$-0.749190\pi$
$613$	− 869.117i	− 1.41781i	0.705304	−	0.708905i	$-0.250810\pi$
$614$	0	0
$615$	− 454.995i	− 0.739829i
$616$	0	0
$617$	−249.359	−0.404147	−0.202074	−	0.979370i	$-0.564768\pi$
$617$	−249.359	−0.404147	−0.202074	+	0.979370i	$0.564768\pi$
$618$	0	0
$619$	−497.673	−0.803995	−0.401998	−	0.915641i	$-0.631684\pi$
$619$	−497.673	−0.803995	−0.401998	+	0.915641i	$0.631684\pi$
$620$	0	0
$621$	− 764.031i	− 1.23032i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−502.486	−0.803977
$626$	0	0
$627$	−61.7490	−0.0984833
$628$	0	0
$629$	179.527i	0.285417i
$630$	0	0
$631$	− 172.763i	− 0.273792i	−0.990585	−	0.136896i	$-0.956287\pi$
$631$	− 172.763i	− 0.273792i	0.990585	−	0.136896i	$-0.0437126\pi$
$632$	0	0
$633$	−316.158	−0.499460
$634$	0	0
$635$	−1016.42	−1.60066
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 81.6300i	− 0.127746i
$640$	0	0
$641$	−427.898	−0.667547	−0.333774	−	0.942653i	$-0.608322\pi$
$641$	−427.898	−0.667547	−0.333774	+	0.942653i	$0.608322\pi$
$642$	0	0
$643$	−15.9463	−0.0247998	−0.0123999	−	0.999923i	$-0.503947\pi$
$643$	−15.9463	−0.0247998	−0.0123999	+	0.999923i	$0.503947\pi$
$644$	0	0
$645$	293.431i	0.454932i
$646$	0	0
$647$	− 520.204i	− 0.804025i	−0.915634	−	0.402012i	$-0.868311\pi$
$647$	− 520.204i	− 0.804025i	0.915634	−	0.402012i	$-0.131689\pi$
$648$	0	0
$649$	145.308	0.223896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 424.568i	− 0.650181i	−0.945683	−	0.325091i	$-0.894605\pi$
$653$	− 424.568i	− 0.650181i	0.945683	−	0.325091i	$-0.105395\pi$
$654$	0	0
$655$	− 1081.23i	− 1.65073i
$656$	0	0
$657$	−619.246	−0.942535
$658$	0	0
$659$	−304.044	−0.461372	−0.230686	−	0.973028i	$-0.574097\pi$
$659$	−304.044	−0.461372	−0.230686	+	0.973028i	$0.574097\pi$
$660$	0	0
$661$	179.108i	0.270966i	0.990780	+	0.135483i	$0.0432586\pi$
$661$	179.108i	0.270966i	−0.990780	+	0.135483i	$0.956741\pi$
$662$	0	0
$663$	394.694i	0.595316i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−690.474	−1.03519
$668$	0	0
$669$	− 67.9739i	− 0.101605i
$670$	0	0
$671$	1348.20i	2.00923i
$672$	0	0
$673$	544.352	0.808844	0.404422	−	0.914573i	$-0.367473\pi$
$673$	544.352	0.808844	0.404422	+	0.914573i	$0.367473\pi$
$674$	0	0
$675$	−105.728	−0.156634
$676$	0	0
$677$	544.344i	0.804053i	0.915628	+	0.402027i	$0.131694\pi$
$677$	544.344i	0.804053i	−0.915628	+	0.402027i	$0.868306\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	116.146	0.170552
$682$	0	0
$683$	−886.988	−1.29866	−0.649332	−	0.760505i	$-0.724951\pi$
$683$	−886.988	−1.29866	−0.649332	+	0.760505i	$0.724951\pi$
$684$	0	0
$685$	44.0730i	0.0643401i
$686$	0	0
$687$	155.841i	0.226842i
$688$	0	0
$689$	87.5704	0.127098
$690$	0	0
$691$	1178.48	1.70548	0.852738	−	0.522340i	$-0.174941\pi$
$691$	1178.48	1.70548	0.852738	+	0.522340i	$0.174941\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	287.633i	0.413861i
$696$	0	0
$697$	788.254	1.13092
$698$	0	0
$699$	−224.706	−0.321468
$700$	0	0
$701$	− 901.601i	− 1.28616i	−0.765797	−	0.643082i	$-0.777655\pi$
$701$	− 901.601i	− 1.28616i	0.765797	−	0.643082i	$-0.222345\pi$
$702$	0	0
$703$	− 41.6868i	− 0.0592984i
$704$	0	0
$705$	−152.932	−0.216925
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 332.802i	− 0.469397i	−0.972068	−	0.234698i	$-0.924590\pi$
$709$	− 332.802i	− 0.469397i	0.972068	−	0.234698i	$-0.0754102\pi$
$710$	0	0
$711$	824.292i	1.15934i
$712$	0	0
$713$	684.126	0.959504
$714$	0	0
$715$	1037.50	1.45104
$716$	0	0
$717$	285.618i	0.398351i
$718$	0	0
$719$	− 1184.95i	− 1.64805i	−0.566551	−	0.824027i	$-0.691723\pi$
$719$	− 1184.95i	− 1.64805i	0.566551	−	0.824027i	$-0.308277\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	542.320	0.750097
$724$	0	0
$725$	95.5488i	0.131791i
$726$	0	0
$727$	− 19.9398i	− 0.0274275i	−0.999906	−	0.0137138i	$-0.995635\pi$
$727$	− 19.9398i	− 0.0274275i	0.999906	−	0.0137138i	$-0.00436536\pi$
$728$	0	0
$729$	279.711	0.383691
$730$	0	0
$731$	−508.354	−0.695423
$732$	0	0
$733$	− 1037.30i	− 1.41514i	−0.706644	−	0.707569i	$-0.749792\pi$
$733$	− 1037.30i	− 1.41514i	0.706644	−	0.707569i	$-0.250208\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	980.462	1.33034
$738$	0	0
$739$	419.971	0.568297	0.284148	−	0.958780i	$-0.408289\pi$
$739$	419.971	0.568297	0.284148	+	0.958780i	$0.408289\pi$
$740$	0	0
$741$	− 91.6491i	− 0.123683i
$742$	0	0
$743$	− 1241.67i	− 1.67116i	−0.549370	−	0.835579i	$-0.685132\pi$
$743$	− 1241.67i	− 1.67116i	0.549370	−	0.835579i	$-0.314868\pi$
$744$	0	0
$745$	1231.10	1.65249
$746$	0	0
$747$	−178.227	−0.238591
$748$	0	0
$749$	0	0
$750$	0	0
$751$	647.797i	0.862579i	0.902214	+	0.431289i	$0.141941\pi$
$751$	647.797i	0.862579i	−0.902214	+	0.431289i	$0.858059\pi$
$752$	0	0
$753$	264.225	0.350896
$754$	0	0
$755$	493.411	0.653524
$756$	0	0
$757$	105.310i	0.139116i	0.997578	+	0.0695578i	$0.0221588\pi$
$757$	105.310i	0.139116i	−0.997578	+	0.0695578i	$0.977841\pi$
$758$	0	0
$759$	618.946i	0.815476i
$760$	0	0
$761$	421.884	0.554381	0.277190	−	0.960815i	$-0.410597\pi$
$761$	421.884	0.554381	0.277190	+	0.960815i	$0.410597\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	373.275i	0.487941i
$766$	0	0
$767$	215.670i	0.281186i
$768$	0	0
$769$	189.767	0.246772	0.123386	−	0.992359i	$-0.460625\pi$
$769$	189.767	0.246772	0.123386	+	0.992359i	$0.460625\pi$
$770$	0	0
$771$	−239.801	−0.311025
$772$	0	0
$773$	842.788i	1.09028i	0.838345	+	0.545141i	$0.183524\pi$
$773$	842.788i	1.09028i	−0.838345	+	0.545141i	$0.816476\pi$
$774$	0	0
$775$	− 94.6704i	− 0.122155i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−183.035	−0.234961
$780$	0	0
$781$	160.876i	0.205988i
$782$	0	0
$783$	− 573.712i	− 0.732710i
$784$	0	0
$785$	−540.181	−0.688128
$786$	0	0
$787$	1494.20	1.89860	0.949298	−	0.314376i	$-0.101795\pi$
$787$	1494.20	1.89860	0.949298	+	0.314376i	$0.101795\pi$
$788$	0	0
$789$	− 334.347i	− 0.423760i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2001.02	−2.52335
$794$	0	0
$795$	−35.8412	−0.0450833
$796$	0	0
$797$	292.040i	0.366424i	0.983073	+	0.183212i	$0.0586494\pi$
$797$	292.040i	0.366424i	−0.983073	+	0.183212i	$0.941351\pi$
$798$	0	0
$799$	− 264.947i	− 0.331598i
$800$	0	0
$801$	−988.827	−1.23449
$802$	0	0
$803$	1220.41	1.51981
$804$	0	0
$805$	0	0
$806$	0	0
$807$	364.893i	0.452160i
$808$	0	0
$809$	156.906	0.193951	0.0969754	−	0.995287i	$-0.469083\pi$
$809$	156.906	0.193951	0.0969754	+	0.995287i	$0.469083\pi$
$810$	0	0
$811$	1183.00	1.45869	0.729347	−	0.684144i	$-0.239824\pi$
$811$	1183.00	1.45869	0.729347	+	0.684144i	$0.239824\pi$
$812$	0	0
$813$	193.527i	0.238040i
$814$	0	0
$815$	− 374.141i	− 0.459068i
$816$	0	0
$817$	118.041	0.144481
$818$	0	0
$819$	0	0
$820$	0	0
$821$	185.415i	0.225840i	0.993604	+	0.112920i	$0.0360204\pi$
$821$	185.415i	0.225840i	−0.993604	+	0.112920i	$0.963980\pi$
$822$	0	0
$823$	− 652.316i	− 0.792608i	−0.918119	−	0.396304i	$-0.870293\pi$
$823$	− 652.316i	− 0.792608i	0.918119	−	0.396304i	$-0.129707\pi$
$824$	0	0
$825$	85.6506	0.103819
$826$	0	0
$827$	829.430	1.00294	0.501469	−	0.865176i	$-0.332793\pi$
$827$	829.430	1.00294	0.501469	+	0.865176i	$0.332793\pi$
$828$	0	0
$829$	755.129i	0.910892i	0.890263	+	0.455446i	$0.150520\pi$
$829$	755.129i	0.910892i	−0.890263	+	0.455446i	$0.849480\pi$
$830$	0	0
$831$	421.277i	0.506952i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	600.226	0.718834
$836$	0	0
$837$	568.437i	0.679137i
$838$	0	0
$839$	88.6147i	0.105619i	0.998605	+	0.0528097i	$0.0168177\pi$
$839$	88.6147i	0.105619i	−0.998605	+	0.0528097i	$0.983182\pi$
$840$	0	0
$841$	322.522	0.383499
$842$	0	0
$843$	−458.366	−0.543731
$844$	0	0
$845$	769.045i	0.910113i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	93.4685	0.110092
$850$	0	0
$851$	−417.851	−0.491011
$852$	0	0
$853$	− 367.466i	− 0.430792i	−0.976527	−	0.215396i	$-0.930896\pi$
$853$	− 367.466i	− 0.430792i	0.976527	−	0.215396i	$-0.0691042\pi$
$854$	0	0
$855$	− 86.6755i	− 0.101375i
$856$	0	0
$857$	165.340	0.192928	0.0964642	−	0.995336i	$-0.469247\pi$
$857$	165.340	0.192928	0.0964642	+	0.995336i	$0.469247\pi$
$858$	0	0
$859$	−650.972	−0.757825	−0.378913	−	0.925432i	$-0.623702\pi$
$859$	−650.972	−0.757825	−0.378913	+	0.925432i	$0.623702\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 136.736i	− 0.158442i	−0.996857	−	0.0792212i	$-0.974757\pi$
$863$	− 136.736i	− 0.158442i	0.996857	−	0.0792212i	$-0.0252433\pi$
$864$	0	0
$865$	502.240	0.580625
$866$	0	0
$867$	−196.633	−0.226797
$868$	0	0
$869$	− 1624.51i	− 1.86941i
$870$	0	0
$871$	1455.22i	1.67075i
$872$	0	0
$873$	−248.859	−0.285062
$874$	0	0
$875$	0	0
$876$	0	0
$877$	176.502i	0.201256i	0.994924	+	0.100628i	$0.0320852\pi$
$877$	176.502i	0.201256i	−0.994924	+	0.100628i	$0.967915\pi$
$878$	0	0
$879$	− 474.634i	− 0.539971i
$880$	0	0
$881$	−734.879	−0.834142	−0.417071	−	0.908874i	$-0.636943\pi$
$881$	−734.879	−0.834142	−0.417071	+	0.908874i	$0.636943\pi$
$882$	0	0
$883$	−872.637	−0.988264	−0.494132	−	0.869387i	$-0.664514\pi$
$883$	−872.637	−0.988264	−0.494132	+	0.869387i	$0.664514\pi$
$884$	0	0
$885$	− 88.2703i	− 0.0997405i
$886$	0	0
$887$	24.0981i	0.0271681i	0.999908	+	0.0135840i	$0.00432407\pi$
$887$	24.0981i	0.0271681i	−0.999908	+	0.0135840i	$0.995676\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	185.590	0.208294
$892$	0	0
$893$	61.5213i	0.0688929i
$894$	0	0
$895$	701.740i	0.784067i
$896$	0	0
$897$	−918.652	−1.02414
$898$	0	0
$899$	513.711	0.571425
$900$	0	0
$901$	− 62.0930i	− 0.0689156i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1037.70	1.14663
$906$	0	0
$907$	1413.31	1.55822	0.779110	−	0.626887i	$-0.215671\pi$
$907$	1413.31	1.55822	0.779110	+	0.626887i	$0.215671\pi$
$908$	0	0
$909$	− 273.990i	− 0.301419i
$910$	0	0
$911$	1778.28i	1.95201i	0.217746	+	0.976005i	$0.430129\pi$
$911$	1778.28i	1.95201i	−0.217746	+	0.976005i	$0.569871\pi$
$912$	0	0
$913$	351.251	0.384722
$914$	0	0
$915$	818.987	0.895067
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 893.841i	− 0.972624i	−0.873785	−	0.486312i	$-0.838342\pi$
$919$	− 893.841i	− 0.972624i	0.873785	−	0.486312i	$-0.161658\pi$
$920$	0	0
$921$	88.4540	0.0960413
$922$	0	0
$923$	−238.776	−0.258696
$924$	0	0
$925$	57.8228i	0.0625111i
$926$	0	0
$927$	− 395.122i	− 0.426237i
$928$	0	0
$929$	729.460	0.785210	0.392605	−	0.919707i	$-0.371574\pi$
$929$	729.460	0.785210	0.392605	+	0.919707i	$0.371574\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 173.300i	− 0.185745i
$934$	0	0
$935$	− 735.651i	− 0.786792i
$936$	0	0
$937$	−637.240	−0.680085	−0.340042	−	0.940410i	$-0.610441\pi$
$937$	−637.240	−0.680085	−0.340042	+	0.940410i	$0.610441\pi$
$938$	0	0
$939$	−347.051	−0.369596
$940$	0	0
$941$	− 1753.57i	− 1.86352i	−0.363072	−	0.931761i	$-0.618272\pi$
$941$	− 1753.57i	− 1.86352i	0.363072	−	0.931761i	$-0.381728\pi$
$942$	0	0
$943$	1834.66i	1.94556i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	873.993	0.922907	0.461453	−	0.887164i	$-0.347328\pi$
$947$	873.993	0.922907	0.461453	+	0.887164i	$0.347328\pi$
$948$	0	0
$949$	1811.36i	1.90870i
$950$	0	0
$951$	103.688i	0.109031i
$952$	0	0
$953$	−880.208	−0.923618	−0.461809	−	0.886979i	$-0.652800\pi$
$953$	−880.208	−0.923618	−0.461809	+	0.886979i	$0.652800\pi$
$954$	0	0
$955$	554.383	0.580506
$956$	0	0
$957$	464.767i	0.485650i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	452.012	0.470356
$962$	0	0
$963$	277.812	0.288486
$964$	0	0
$965$	− 389.771i	− 0.403908i
$966$	0	0
$967$	259.016i	0.267855i	0.990991	+	0.133928i	$0.0427590\pi$
$967$	259.016i	0.267855i	−0.990991	+	0.133928i	$0.957241\pi$
$968$	0	0
$969$	−64.9851	−0.0670640
$970$	0	0
$971$	−432.650	−0.445572	−0.222786	−	0.974867i	$-0.571515\pi$
$971$	−432.650	−0.445572	−0.222786	+	0.974867i	$0.571515\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	127.124i	0.130384i
$976$	0	0
$977$	815.509	0.834708	0.417354	−	0.908744i	$-0.362958\pi$
$977$	815.509	0.834708	0.417354	+	0.908744i	$0.362958\pi$
$978$	0	0
$979$	1948.78	1.99058
$980$	0	0
$981$	55.3730i	0.0564455i
$982$	0	0
$983$	− 139.412i	− 0.141823i	−0.997483	−	0.0709116i	$-0.977409\pi$
$983$	− 139.412i	− 0.141823i	0.997483	−	0.0709116i	$-0.0225908\pi$
$984$	0	0
$985$	−976.533	−0.991404
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1183.20i	− 1.19636i
$990$	0	0
$991$	1548.78i	1.56285i	0.624002	+	0.781423i	$0.285506\pi$
$991$	1548.78i	1.56285i	−0.624002	+	0.781423i	$0.714494\pi$
$992$	0	0
$993$	−324.114	−0.326399
$994$	0	0
$995$	−1132.18	−1.13786
$996$	0	0
$997$	− 1282.50i	− 1.28636i	−0.765717	−	0.643178i	$-0.777616\pi$
$997$	− 1282.50i	− 1.28636i	0.765717	−	0.643178i	$-0.222384\pi$
$998$	0	0
$999$	− 347.190i	− 0.347538i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.g.l.687.3		6
4.3	odd	2		392.3.g.i.99.2		6
7.3	odd	6		224.3.o.d.79.3		12
7.5	odd	6		224.3.o.d.207.4		12
7.6	odd	2		1568.3.g.j.687.4		6
8.3	odd	2	inner	1568.3.g.l.687.4		6
8.5	even	2		392.3.g.i.99.1		6
28.3	even	6		56.3.k.d.51.4	yes	12
28.11	odd	6		392.3.k.l.275.4		12
28.19	even	6		56.3.k.d.11.5	yes	12
28.23	odd	6		392.3.k.l.67.5		12
28.27	even	2		392.3.g.j.99.2		6
56.3	even	6		224.3.o.d.79.4		12
56.5	odd	6		56.3.k.d.11.4	✓	12
56.13	odd	2		392.3.g.j.99.1		6
56.19	even	6		224.3.o.d.207.3		12
56.27	even	2		1568.3.g.j.687.3		6
56.37	even	6		392.3.k.l.67.4		12
56.45	odd	6		56.3.k.d.51.5	yes	12
56.53	even	6		392.3.k.l.275.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.k.d.11.4	✓	12	56.5	odd	6
56.3.k.d.11.5	yes	12	28.19	even	6
56.3.k.d.51.4	yes	12	28.3	even	6
56.3.k.d.51.5	yes	12	56.45	odd	6
224.3.o.d.79.3		12	7.3	odd	6
224.3.o.d.79.4		12	56.3	even	6
224.3.o.d.207.3		12	56.19	even	6
224.3.o.d.207.4		12	7.5	odd	6
392.3.g.i.99.1		6	8.5	even	2
392.3.g.i.99.2		6	4.3	odd	2
392.3.g.j.99.1		6	56.13	odd	2
392.3.g.j.99.2		6	28.27	even	2
392.3.k.l.67.4		12	56.37	even	6
392.3.k.l.67.5		12	28.23	odd	6
392.3.k.l.275.4		12	28.11	odd	6
392.3.k.l.275.5		12	56.53	even	6
1568.3.g.j.687.3		6	56.27	even	2
1568.3.g.j.687.4		6	7.6	odd	2
1568.3.g.l.687.3		6	1.1	even	1	trivial
1568.3.g.l.687.4		6	8.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.64878 q^{3} -4.56111i q^{5} -6.28154 q^{9} +O(q^{10})\)	\(q+1.64878 q^{3} -4.56111i q^{5} -6.28154 q^{9} +12.3797 q^{11} +18.3741i q^{13} -7.52026i q^{15} +13.0284 q^{17} -3.02524 q^{19} +30.3237i q^{23} +4.19624 q^{25} -25.1958 q^{27} +22.7701i q^{29} -22.5608i q^{31} +20.4113 q^{33} +13.7797i q^{37} +30.2948i q^{39} +60.5026 q^{41} -39.0188 q^{43} +28.6508i q^{45} -20.3360i q^{47} +21.4810 q^{51} -4.76596i q^{53} -56.4650i q^{55} -4.98794 q^{57} +11.7377 q^{59} +108.904i q^{61} +83.8066 q^{65} +79.1994 q^{67} +49.9970i q^{69} +12.9952i q^{71} +98.5819 q^{73} +6.91866 q^{75} -131.225i q^{79} +14.9915 q^{81} +28.3732 q^{83} -59.4242i q^{85} +37.5428i q^{87} +157.418 q^{89} -37.1977i q^{93} +13.7985i q^{95} +39.6175 q^{97} -77.7633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 40 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 40 * q^9	\( 6 q + 6 q^{3} + 40 q^{9} + 30 q^{11} + 30 q^{17} - 78 q^{19} + 92 q^{25} + 78 q^{27} - 78 q^{33} + 116 q^{41} + 100 q^{43} + 10 q^{51} + 166 q^{57} + 110 q^{59} + 32 q^{65} + 434 q^{67} + 102 q^{73} + 60 q^{75} + 82 q^{81} - 268 q^{83} + 214 q^{89} + 76 q^{97} - 252 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 40 * q^9 + 30 * q^11 + 30 * q^17 - 78 * q^19 + 92 * q^25 + 78 * q^27 - 78 * q^33 + 116 * q^41 + 100 * q^43 + 10 * q^51 + 166 * q^57 + 110 * q^59 + 32 * q^65 + 434 * q^67 + 102 * q^73 + 60 * q^75 + 82 * q^81 - 268 * q^83 + 214 * q^89 + 76 * q^97 - 252 * q^99

Introduction

L-functions

Modular forms

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