LMFDB - Embedded newform 4560.2.a.bo.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4560,2,Mod(1,4560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4560.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,2,0,2,0,2,0,-6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.4117833217$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.64575 q^{7} +1.00000 q^{9} -0.354249 q^{11} -0.354249 q^{13} +1.00000 q^{15} -4.00000 q^{17} +1.00000 q^{19} -1.64575 q^{21} -9.29150 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.93725 q^{29} -6.00000 q^{31} -0.354249 q^{33} -1.64575 q^{35} +3.64575 q^{37} -0.354249 q^{39} -9.64575 q^{41} -5.64575 q^{43} +1.00000 q^{45} +1.29150 q^{47} -4.29150 q^{49} -4.00000 q^{51} +11.2915 q^{53} -0.354249 q^{55} +1.00000 q^{57} -11.2915 q^{59} -11.2915 q^{61} -1.64575 q^{63} -0.354249 q^{65} +6.58301 q^{67} -9.29150 q^{69} -7.29150 q^{71} +10.0000 q^{73} +1.00000 q^{75} +0.583005 q^{77} -6.58301 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} +8.93725 q^{87} -16.9373 q^{89} +0.583005 q^{91} -6.00000 q^{93} +1.00000 q^{95} -2.93725 q^{97} -0.354249 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^17 + 2 * q^19 + 2 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 - 12 * q^31 - 6 * q^33 + 2 * q^35 + 2 * q^37 - 6 * q^39 - 14 * q^41 - 6 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 8 * q^51 + 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 12 * q^61 + 2 * q^63 - 6 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 20 * q^73 + 2 * q^75 - 20 * q^77 + 8 * q^79 + 2 * q^81 - 12 * q^83 - 8 * q^85 + 2 * q^87 - 18 * q^89 - 20 * q^91 - 12 * q^93 + 2 * q^95 + 10 * q^97 - 6 * q^99

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	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.64575	−0.622036	−0.311018	−	0.950404i	$-0.600670\pi$
$7$	−1.64575	−0.622036	−0.311018	+	0.950404i	$0.600670\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.354249	−0.106810	−0.0534050	−	0.998573i	$-0.517007\pi$
$11$	−0.354249	−0.106810	−0.0534050	+	0.998573i	$0.517007\pi$
$12$	0	0
$13$	−0.354249	−0.0982509	−0.0491255	−	0.998793i	$-0.515643\pi$
$13$	−0.354249	−0.0982509	−0.0491255	+	0.998793i	$0.515643\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.64575	−0.359132
$22$	0	0
$23$	−9.29150	−1.93741	−0.968706	−	0.248211i	$-0.920158\pi$
$23$	−9.29150	−1.93741	−0.968706	+	0.248211i	$0.920158\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.93725	1.65961	0.829803	−	0.558056i	$-0.188453\pi$
$29$	8.93725	1.65961	0.829803	+	0.558056i	$0.188453\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−0.354249	−0.0616668
$34$	0	0
$35$	−1.64575	−0.278183
$36$	0	0
$37$	3.64575	0.599358	0.299679	−	0.954040i	$-0.403120\pi$
$37$	3.64575	0.599358	0.299679	+	0.954040i	$0.403120\pi$
$38$	0	0
$39$	−0.354249	−0.0567252
$40$	0	0
$41$	−9.64575	−1.50641	−0.753207	−	0.657784i	$-0.771494\pi$
$41$	−9.64575	−1.50641	−0.753207	+	0.657784i	$0.771494\pi$
$42$	0	0
$43$	−5.64575	−0.860969	−0.430485	−	0.902598i	$-0.641657\pi$
$43$	−5.64575	−0.860969	−0.430485	+	0.902598i	$0.641657\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	1.29150	0.188385	0.0941925	−	0.995554i	$-0.469973\pi$
$47$	1.29150	0.188385	0.0941925	+	0.995554i	$0.469973\pi$
$48$	0	0
$49$	−4.29150	−0.613072
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	11.2915	1.55101	0.775504	−	0.631343i	$-0.217496\pi$
$53$	11.2915	1.55101	0.775504	+	0.631343i	$0.217496\pi$
$54$	0	0
$55$	−0.354249	−0.0477669
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−11.2915	−1.47003	−0.735014	−	0.678052i	$-0.762825\pi$
$59$	−11.2915	−1.47003	−0.735014	+	0.678052i	$0.762825\pi$
$60$	0	0
$61$	−11.2915	−1.44573	−0.722864	−	0.690990i	$-0.757175\pi$
$61$	−11.2915	−1.44573	−0.722864	+	0.690990i	$0.757175\pi$
$62$	0	0
$63$	−1.64575	−0.207345
$64$	0	0
$65$	−0.354249	−0.0439391
$66$	0	0
$67$	6.58301	0.804242	0.402121	−	0.915587i	$-0.368273\pi$
$67$	6.58301	0.804242	0.402121	+	0.915587i	$0.368273\pi$
$68$	0	0
$69$	−9.29150	−1.11857
$70$	0	0
$71$	−7.29150	−0.865342	−0.432671	−	0.901552i	$-0.642429\pi$
$71$	−7.29150	−0.865342	−0.432671	+	0.901552i	$0.642429\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0.583005	0.0664396
$78$	0	0
$79$	−6.58301	−0.740646	−0.370323	−	0.928903i	$-0.620753\pi$
$79$	−6.58301	−0.740646	−0.370323	+	0.928903i	$0.620753\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	8.93725	0.958174
$88$	0	0
$89$	−16.9373	−1.79535	−0.897673	−	0.440663i	$-0.854743\pi$
$89$	−16.9373	−1.79535	−0.897673	+	0.440663i	$0.854743\pi$
$90$	0	0
$91$	0.583005	0.0611156
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−2.93725	−0.298233	−0.149116	−	0.988820i	$-0.547643\pi$
$97$	−2.93725	−0.298233	−0.149116	+	0.988820i	$0.547643\pi$
$98$	0	0
$99$	−0.354249	−0.0356033
$100$	0	0
$101$	9.29150	0.924539	0.462270	−	0.886739i	$-0.347035\pi$
$101$	9.29150	0.924539	0.462270	+	0.886739i	$0.347035\pi$
$102$	0	0
$103$	10.5830	1.04277	0.521387	−	0.853320i	$-0.325415\pi$
$103$	10.5830	1.04277	0.521387	+	0.853320i	$0.325415\pi$
$104$	0	0
$105$	−1.64575	−0.160609
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	5.29150	0.506834	0.253417	−	0.967357i	$-0.418446\pi$
$109$	5.29150	0.506834	0.253417	+	0.967357i	$0.418446\pi$
$110$	0	0
$111$	3.64575	0.346039
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	−9.29150	−0.866437
$116$	0	0
$117$	−0.354249	−0.0327503
$118$	0	0
$119$	6.58301	0.603463
$120$	0	0
$121$	−10.8745	−0.988592
$122$	0	0
$123$	−9.64575	−0.869728
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−5.64575	−0.497081
$130$	0	0
$131$	14.2288	1.24317	0.621586	−	0.783346i	$-0.286489\pi$
$131$	14.2288	1.24317	0.621586	+	0.783346i	$0.286489\pi$
$132$	0	0
$133$	−1.64575	−0.142705
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−17.8745	−1.51610	−0.758048	−	0.652199i	$-0.773847\pi$
$139$	−17.8745	−1.51610	−0.758048	+	0.652199i	$0.773847\pi$
$140$	0	0
$141$	1.29150	0.108764
$142$	0	0
$143$	0.125492	0.0104942
$144$	0	0
$145$	8.93725	0.742199
$146$	0	0
$147$	−4.29150	−0.353957
$148$	0	0
$149$	−20.5830	−1.68623	−0.843113	−	0.537737i	$-0.819279\pi$
$149$	−20.5830	−1.68623	−0.843113	+	0.537737i	$0.819279\pi$
$150$	0	0
$151$	8.58301	0.698475	0.349238	−	0.937034i	$-0.386441\pi$
$151$	8.58301	0.698475	0.349238	+	0.937034i	$0.386441\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−13.2915	−1.06078	−0.530389	−	0.847755i	$-0.677954\pi$
$157$	−13.2915	−1.06078	−0.530389	+	0.847755i	$0.677954\pi$
$158$	0	0
$159$	11.2915	0.895474
$160$	0	0
$161$	15.2915	1.20514
$162$	0	0
$163$	2.35425	0.184399	0.0921995	−	0.995741i	$-0.470610\pi$
$163$	2.35425	0.184399	0.0921995	+	0.995741i	$0.470610\pi$
$164$	0	0
$165$	−0.354249	−0.0275782
$166$	0	0
$167$	21.2915	1.64759	0.823793	−	0.566891i	$-0.191854\pi$
$167$	21.2915	1.64759	0.823793	+	0.566891i	$0.191854\pi$
$168$	0	0
$169$	−12.8745	−0.990347
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−1.64575	−0.124407
$176$	0	0
$177$	−11.2915	−0.848721
$178$	0	0
$179$	7.29150	0.544992	0.272496	−	0.962157i	$-0.412151\pi$
$179$	7.29150	0.544992	0.272496	+	0.962157i	$0.412151\pi$
$180$	0	0
$181$	17.2915	1.28527	0.642634	−	0.766174i	$-0.277842\pi$
$181$	17.2915	1.28527	0.642634	+	0.766174i	$0.277842\pi$
$182$	0	0
$183$	−11.2915	−0.834692
$184$	0	0
$185$	3.64575	0.268041
$186$	0	0
$187$	1.41699	0.103621
$188$	0	0
$189$	−1.64575	−0.119711
$190$	0	0
$191$	−6.22876	−0.450697	−0.225349	−	0.974278i	$-0.572352\pi$
$191$	−6.22876	−0.450697	−0.225349	+	0.974278i	$0.572352\pi$
$192$	0	0
$193$	−11.6458	−0.838280	−0.419140	−	0.907922i	$-0.637668\pi$
$193$	−11.6458	−0.838280	−0.419140	+	0.907922i	$0.637668\pi$
$194$	0	0
$195$	−0.354249	−0.0253683
$196$	0	0
$197$	−25.1660	−1.79300	−0.896502	−	0.443040i	$-0.853900\pi$
$197$	−25.1660	−1.79300	−0.896502	+	0.443040i	$0.853900\pi$
$198$	0	0
$199$	−7.29150	−0.516881	−0.258440	−	0.966027i	$-0.583209\pi$
$199$	−7.29150	−0.516881	−0.258440	+	0.966027i	$0.583209\pi$
$200$	0	0
$201$	6.58301	0.464329
$202$	0	0
$203$	−14.7085	−1.03233
$204$	0	0
$205$	−9.64575	−0.673688
$206$	0	0
$207$	−9.29150	−0.645804
$208$	0	0
$209$	−0.354249	−0.0245039
$210$	0	0
$211$	−2.58301	−0.177821	−0.0889107	−	0.996040i	$-0.528339\pi$
$211$	−2.58301	−0.177821	−0.0889107	+	0.996040i	$0.528339\pi$
$212$	0	0
$213$	−7.29150	−0.499606
$214$	0	0
$215$	−5.64575	−0.385037
$216$	0	0
$217$	9.87451	0.670325
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	1.41699	0.0953174
$222$	0	0
$223$	−2.58301	−0.172971	−0.0864854	−	0.996253i	$-0.527564\pi$
$223$	−2.58301	−0.172971	−0.0864854	+	0.996253i	$0.527564\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.7085	0.710748	0.355374	−	0.934724i	$-0.384354\pi$
$227$	10.7085	0.710748	0.355374	+	0.934724i	$0.384354\pi$
$228$	0	0
$229$	8.70850	0.575474	0.287737	−	0.957710i	$-0.407097\pi$
$229$	8.70850	0.575474	0.287737	+	0.957710i	$0.407097\pi$
$230$	0	0
$231$	0.583005	0.0383589
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	1.29150	0.0842483
$236$	0	0
$237$	−6.58301	−0.427612
$238$	0	0
$239$	−15.6458	−1.01204	−0.506020	−	0.862522i	$-0.668884\pi$
$239$	−15.6458	−1.01204	−0.506020	+	0.862522i	$0.668884\pi$
$240$	0	0
$241$	16.5830	1.06821	0.534103	−	0.845420i	$-0.320650\pi$
$241$	16.5830	1.06821	0.534103	+	0.845420i	$0.320650\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.29150	−0.274174
$246$	0	0
$247$	−0.354249	−0.0225403
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	16.3542	1.03227	0.516136	−	0.856507i	$-0.327370\pi$
$251$	16.3542	1.03227	0.516136	+	0.856507i	$0.327370\pi$
$252$	0	0
$253$	3.29150	0.206935
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−5.41699	−0.337903	−0.168951	−	0.985624i	$-0.554038\pi$
$257$	−5.41699	−0.337903	−0.168951	+	0.985624i	$0.554038\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	8.93725	0.553202
$262$	0	0
$263$	−4.58301	−0.282600	−0.141300	−	0.989967i	$-0.545128\pi$
$263$	−4.58301	−0.282600	−0.141300	+	0.989967i	$0.545128\pi$
$264$	0	0
$265$	11.2915	0.693631
$266$	0	0
$267$	−16.9373	−1.03654
$268$	0	0
$269$	4.22876	0.257832	0.128916	−	0.991656i	$-0.458850\pi$
$269$	4.22876	0.257832	0.128916	+	0.991656i	$0.458850\pi$
$270$	0	0
$271$	4.70850	0.286021	0.143010	−	0.989721i	$-0.454322\pi$
$271$	4.70850	0.286021	0.143010	+	0.989721i	$0.454322\pi$
$272$	0	0
$273$	0.583005	0.0352851
$274$	0	0
$275$	−0.354249	−0.0213620
$276$	0	0
$277$	0.583005	0.0350294	0.0175147	−	0.999847i	$-0.494425\pi$
$277$	0.583005	0.0350294	0.0175147	+	0.999847i	$0.494425\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−32.2288	−1.92261	−0.961303	−	0.275493i	$-0.911159\pi$
$281$	−32.2288	−1.92261	−0.961303	+	0.275493i	$0.911159\pi$
$282$	0	0
$283$	11.5203	0.684808	0.342404	−	0.939553i	$-0.388759\pi$
$283$	11.5203	0.684808	0.342404	+	0.939553i	$0.388759\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	15.8745	0.937043
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−2.93725	−0.172185
$292$	0	0
$293$	5.41699	0.316464	0.158232	−	0.987402i	$-0.449421\pi$
$293$	5.41699	0.316464	0.158232	+	0.987402i	$0.449421\pi$
$294$	0	0
$295$	−11.2915	−0.657417
$296$	0	0
$297$	−0.354249	−0.0205556
$298$	0	0
$299$	3.29150	0.190353
$300$	0	0
$301$	9.29150	0.535553
$302$	0	0
$303$	9.29150	0.533783
$304$	0	0
$305$	−11.2915	−0.646550
$306$	0	0
$307$	24.4575	1.39586	0.697932	−	0.716164i	$-0.254104\pi$
$307$	24.4575	1.39586	0.697932	+	0.716164i	$0.254104\pi$
$308$	0	0
$309$	10.5830	0.602046
$310$	0	0
$311$	−22.9373	−1.30065	−0.650326	−	0.759655i	$-0.725368\pi$
$311$	−22.9373	−1.30065	−0.650326	+	0.759655i	$0.725368\pi$
$312$	0	0
$313$	17.2915	0.977374	0.488687	−	0.872459i	$-0.337476\pi$
$313$	17.2915	0.977374	0.488687	+	0.872459i	$0.337476\pi$
$314$	0	0
$315$	−1.64575	−0.0927276
$316$	0	0
$317$	20.4575	1.14901	0.574504	−	0.818502i	$-0.305195\pi$
$317$	20.4575	1.14901	0.574504	+	0.818502i	$0.305195\pi$
$318$	0	0
$319$	−3.16601	−0.177263
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−0.354249	−0.0196502
$326$	0	0
$327$	5.29150	0.292621
$328$	0	0
$329$	−2.12549	−0.117182
$330$	0	0
$331$	−11.4170	−0.627535	−0.313767	−	0.949500i	$-0.601591\pi$
$331$	−11.4170	−0.627535	−0.313767	+	0.949500i	$0.601591\pi$
$332$	0	0
$333$	3.64575	0.199786
$334$	0	0
$335$	6.58301	0.359668
$336$	0	0
$337$	14.9373	0.813684	0.406842	−	0.913499i	$-0.366630\pi$
$337$	14.9373	0.813684	0.406842	+	0.913499i	$0.366630\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	2.12549	0.115102
$342$	0	0
$343$	18.5830	1.00339
$344$	0	0
$345$	−9.29150	−0.500238
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	23.1660	1.24005	0.620024	−	0.784583i	$-0.287123\pi$
$349$	23.1660	1.24005	0.620024	+	0.784583i	$0.287123\pi$
$350$	0	0
$351$	−0.354249	−0.0189084
$352$	0	0
$353$	0.583005	0.0310302	0.0155151	−	0.999880i	$-0.495061\pi$
$353$	0.583005	0.0310302	0.0155151	+	0.999880i	$0.495061\pi$
$354$	0	0
$355$	−7.29150	−0.386993
$356$	0	0
$357$	6.58301	0.348410
$358$	0	0
$359$	36.1033	1.90546	0.952729	−	0.303822i	$-0.0982629\pi$
$359$	36.1033	1.90546	0.952729	+	0.303822i	$0.0982629\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−10.8745	−0.570764
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−1.64575	−0.0859075	−0.0429538	−	0.999077i	$-0.513677\pi$
$367$	−1.64575	−0.0859075	−0.0429538	+	0.999077i	$0.513677\pi$
$368$	0	0
$369$	−9.64575	−0.502138
$370$	0	0
$371$	−18.5830	−0.964782
$372$	0	0
$373$	5.06275	0.262139	0.131070	−	0.991373i	$-0.458159\pi$
$373$	5.06275	0.262139	0.131070	+	0.991373i	$0.458159\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.16601	−0.163058
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	18.5830	0.949547	0.474774	−	0.880108i	$-0.342530\pi$
$383$	18.5830	0.949547	0.474774	+	0.880108i	$0.342530\pi$
$384$	0	0
$385$	0.583005	0.0297127
$386$	0	0
$387$	−5.64575	−0.286990
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	37.1660	1.87957
$392$	0	0
$393$	14.2288	0.717746
$394$	0	0
$395$	−6.58301	−0.331227
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	−1.64575	−0.0823906
$400$	0	0
$401$	−4.93725	−0.246555	−0.123277	−	0.992372i	$-0.539340\pi$
$401$	−4.93725	−0.246555	−0.123277	+	0.992372i	$0.539340\pi$
$402$	0	0
$403$	2.12549	0.105878
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−1.29150	−0.0640174
$408$	0	0
$409$	−6.70850	−0.331714	−0.165857	−	0.986150i	$-0.553039\pi$
$409$	−6.70850	−0.331714	−0.165857	+	0.986150i	$0.553039\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	18.5830	0.914410
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−17.8745	−0.875318
$418$	0	0
$419$	−38.9373	−1.90221	−0.951105	−	0.308869i	$-0.900050\pi$
$419$	−38.9373	−1.90221	−0.951105	+	0.308869i	$0.900050\pi$
$420$	0	0
$421$	28.5830	1.39305	0.696525	−	0.717532i	$-0.254728\pi$
$421$	28.5830	1.39305	0.696525	+	0.717532i	$0.254728\pi$
$422$	0	0
$423$	1.29150	0.0627950
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	18.5830	0.899295
$428$	0	0
$429$	0.125492	0.00605882
$430$	0	0
$431$	−3.29150	−0.158546	−0.0792731	−	0.996853i	$-0.525260\pi$
$431$	−3.29150	−0.158546	−0.0792731	+	0.996853i	$0.525260\pi$
$432$	0	0
$433$	−27.6458	−1.32857	−0.664285	−	0.747479i	$-0.731264\pi$
$433$	−27.6458	−1.32857	−0.664285	+	0.747479i	$0.731264\pi$
$434$	0	0
$435$	8.93725	0.428509
$436$	0	0
$437$	−9.29150	−0.444473
$438$	0	0
$439$	1.41699	0.0676295	0.0338147	−	0.999428i	$-0.489234\pi$
$439$	1.41699	0.0676295	0.0338147	+	0.999428i	$0.489234\pi$
$440$	0	0
$441$	−4.29150	−0.204357
$442$	0	0
$443$	−26.7085	−1.26896	−0.634480	−	0.772940i	$-0.718786\pi$
$443$	−26.7085	−1.26896	−0.634480	+	0.772940i	$0.718786\pi$
$444$	0	0
$445$	−16.9373	−0.802903
$446$	0	0
$447$	−20.5830	−0.973543
$448$	0	0
$449$	−36.2288	−1.70974	−0.854870	−	0.518842i	$-0.826363\pi$
$449$	−36.2288	−1.70974	−0.854870	+	0.518842i	$0.826363\pi$
$450$	0	0
$451$	3.41699	0.160900
$452$	0	0
$453$	8.58301	0.403265
$454$	0	0
$455$	0.583005	0.0273317
$456$	0	0
$457$	0.125492	0.00587027	0.00293514	−	0.999996i	$-0.499066\pi$
$457$	0.125492	0.00587027	0.00293514	+	0.999996i	$0.499066\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−37.7490	−1.75815	−0.879073	−	0.476686i	$-0.841838\pi$
$461$	−37.7490	−1.75815	−0.879073	+	0.476686i	$0.841838\pi$
$462$	0	0
$463$	35.5203	1.65077	0.825383	−	0.564573i	$-0.190959\pi$
$463$	35.5203	1.65077	0.825383	+	0.564573i	$0.190959\pi$
$464$	0	0
$465$	−6.00000	−0.278243
$466$	0	0
$467$	−19.8745	−0.919683	−0.459841	−	0.888001i	$-0.652094\pi$
$467$	−19.8745	−0.919683	−0.459841	+	0.888001i	$0.652094\pi$
$468$	0	0
$469$	−10.8340	−0.500267
$470$	0	0
$471$	−13.2915	−0.612440
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	11.2915	0.517002
$478$	0	0
$479$	−4.35425	−0.198951	−0.0994754	−	0.995040i	$-0.531716\pi$
$479$	−4.35425	−0.198951	−0.0994754	+	0.995040i	$0.531716\pi$
$480$	0	0
$481$	−1.29150	−0.0588875
$482$	0	0
$483$	15.2915	0.695787
$484$	0	0
$485$	−2.93725	−0.133374
$486$	0	0
$487$	−17.8745	−0.809971	−0.404986	−	0.914323i	$-0.632723\pi$
$487$	−17.8745	−0.809971	−0.404986	+	0.914323i	$0.632723\pi$
$488$	0	0
$489$	2.35425	0.106463
$490$	0	0
$491$	5.77124	0.260453	0.130226	−	0.991484i	$-0.458430\pi$
$491$	5.77124	0.260453	0.130226	+	0.991484i	$0.458430\pi$
$492$	0	0
$493$	−35.7490	−1.61005
$494$	0	0
$495$	−0.354249	−0.0159223
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	31.0405	1.38956	0.694782	−	0.719220i	$-0.255501\pi$
$499$	31.0405	1.38956	0.694782	+	0.719220i	$0.255501\pi$
$500$	0	0
$501$	21.2915	0.951234
$502$	0	0
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	9.29150	0.413466
$506$	0	0
$507$	−12.8745	−0.571777
$508$	0	0
$509$	−34.1033	−1.51160	−0.755800	−	0.654802i	$-0.772752\pi$
$509$	−34.1033	−1.51160	−0.755800	+	0.654802i	$0.772752\pi$
$510$	0	0
$511$	−16.4575	−0.728038
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	10.5830	0.466343
$516$	0	0
$517$	−0.457513	−0.0201214
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.1033	0.968362	0.484181	−	0.874968i	$-0.339118\pi$
$521$	22.1033	0.968362	0.484181	+	0.874968i	$0.339118\pi$
$522$	0	0
$523$	23.2915	1.01847	0.509233	−	0.860629i	$-0.329929\pi$
$523$	23.2915	1.01847	0.509233	+	0.860629i	$0.329929\pi$
$524$	0	0
$525$	−1.64575	−0.0718265
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	63.3320	2.75357
$530$	0	0
$531$	−11.2915	−0.490009
$532$	0	0
$533$	3.41699	0.148006
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.29150	0.314652
$538$	0	0
$539$	1.52026	0.0654822
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	17.2915	0.742049
$544$	0	0
$545$	5.29150	0.226663
$546$	0	0
$547$	−1.87451	−0.0801482	−0.0400741	−	0.999197i	$-0.512759\pi$
$547$	−1.87451	−0.0801482	−0.0400741	+	0.999197i	$0.512759\pi$
$548$	0	0
$549$	−11.2915	−0.481910
$550$	0	0
$551$	8.93725	0.380740
$552$	0	0
$553$	10.8340	0.460708
$554$	0	0
$555$	3.64575	0.154754
$556$	0	0
$557$	−11.4170	−0.483754	−0.241877	−	0.970307i	$-0.577763\pi$
$557$	−11.4170	−0.483754	−0.241877	+	0.970307i	$0.577763\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	1.41699	0.0598256
$562$	0	0
$563$	−8.12549	−0.342449	−0.171224	−	0.985232i	$-0.554772\pi$
$563$	−8.12549	−0.342449	−0.171224	+	0.985232i	$0.554772\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	0	0
$567$	−1.64575	−0.0691151
$568$	0	0
$569$	−7.06275	−0.296086	−0.148043	−	0.988981i	$-0.547297\pi$
$569$	−7.06275	−0.296086	−0.148043	+	0.988981i	$0.547297\pi$
$570$	0	0
$571$	−16.7085	−0.699229	−0.349614	−	0.936894i	$-0.613687\pi$
$571$	−16.7085	−0.699229	−0.349614	+	0.936894i	$0.613687\pi$
$572$	0	0
$573$	−6.22876	−0.260210
$574$	0	0
$575$	−9.29150	−0.387482
$576$	0	0
$577$	−13.2915	−0.553332	−0.276666	−	0.960966i	$-0.589230\pi$
$577$	−13.2915	−0.553332	−0.276666	+	0.960966i	$0.589230\pi$
$578$	0	0
$579$	−11.6458	−0.483981
$580$	0	0
$581$	9.87451	0.409664
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	−0.354249	−0.0146464
$586$	0	0
$587$	33.2915	1.37409	0.687044	−	0.726616i	$-0.258908\pi$
$587$	33.2915	1.37409	0.687044	+	0.726616i	$0.258908\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	−25.1660	−1.03519
$592$	0	0
$593$	3.41699	0.140319	0.0701596	−	0.997536i	$-0.477649\pi$
$593$	3.41699	0.140319	0.0701596	+	0.997536i	$0.477649\pi$
$594$	0	0
$595$	6.58301	0.269877
$596$	0	0
$597$	−7.29150	−0.298421
$598$	0	0
$599$	−9.41699	−0.384768	−0.192384	−	0.981320i	$-0.561622\pi$
$599$	−9.41699	−0.384768	−0.192384	+	0.981320i	$0.561622\pi$
$600$	0	0
$601$	22.7085	0.926299	0.463149	−	0.886280i	$-0.346719\pi$
$601$	22.7085	0.926299	0.463149	+	0.886280i	$0.346719\pi$
$602$	0	0
$603$	6.58301	0.268081
$604$	0	0
$605$	−10.8745	−0.442112
$606$	0	0
$607$	43.0405	1.74696	0.873480	−	0.486859i	$-0.161858\pi$
$607$	43.0405	1.74696	0.873480	+	0.486859i	$0.161858\pi$
$608$	0	0
$609$	−14.7085	−0.596018
$610$	0	0
$611$	−0.457513	−0.0185090
$612$	0	0
$613$	−30.4575	−1.23017	−0.615084	−	0.788462i	$-0.710878\pi$
$613$	−30.4575	−1.23017	−0.615084	+	0.788462i	$0.710878\pi$
$614$	0	0
$615$	−9.64575	−0.388954
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−23.2915	−0.936165	−0.468082	−	0.883685i	$-0.655055\pi$
$619$	−23.2915	−0.936165	−0.468082	+	0.883685i	$0.655055\pi$
$620$	0	0
$621$	−9.29150	−0.372855
$622$	0	0
$623$	27.8745	1.11677
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.354249	−0.0141473
$628$	0	0
$629$	−14.5830	−0.581462
$630$	0	0
$631$	14.5830	0.580540	0.290270	−	0.956945i	$-0.406255\pi$
$631$	14.5830	0.580540	0.290270	+	0.956945i	$0.406255\pi$
$632$	0	0
$633$	−2.58301	−0.102665
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	1.52026	0.0602349
$638$	0	0
$639$	−7.29150	−0.288447
$640$	0	0
$641$	−16.9373	−0.668981	−0.334491	−	0.942399i	$-0.608564\pi$
$641$	−16.9373	−0.668981	−0.334491	+	0.942399i	$0.608564\pi$
$642$	0	0
$643$	13.6458	0.538136	0.269068	−	0.963121i	$-0.413284\pi$
$643$	13.6458	0.538136	0.269068	+	0.963121i	$0.413284\pi$
$644$	0	0
$645$	−5.64575	−0.222301
$646$	0	0
$647$	3.41699	0.134336	0.0671680	−	0.997742i	$-0.478604\pi$
$647$	3.41699	0.134336	0.0671680	+	0.997742i	$0.478604\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	9.87451	0.387012
$652$	0	0
$653$	9.41699	0.368515	0.184258	−	0.982878i	$-0.441012\pi$
$653$	9.41699	0.368515	0.184258	+	0.982878i	$0.441012\pi$
$654$	0	0
$655$	14.2288	0.555964
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−25.4170	−0.990106	−0.495053	−	0.868863i	$-0.664851\pi$
$659$	−25.4170	−0.990106	−0.495053	+	0.868863i	$0.664851\pi$
$660$	0	0
$661$	1.29150	0.0502336	0.0251168	−	0.999685i	$-0.492004\pi$
$661$	1.29150	0.0502336	0.0251168	+	0.999685i	$0.492004\pi$
$662$	0	0
$663$	1.41699	0.0550315
$664$	0	0
$665$	−1.64575	−0.0638195
$666$	0	0
$667$	−83.0405	−3.21534
$668$	0	0
$669$	−2.58301	−0.0998648
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	−13.0627	−0.503532	−0.251766	−	0.967788i	$-0.581011\pi$
$673$	−13.0627	−0.503532	−0.251766	+	0.967788i	$0.581011\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	40.4575	1.55491	0.777454	−	0.628939i	$-0.216511\pi$
$677$	40.4575	1.55491	0.777454	+	0.628939i	$0.216511\pi$
$678$	0	0
$679$	4.83399	0.185511
$680$	0	0
$681$	10.7085	0.410351
$682$	0	0
$683$	−19.7490	−0.755675	−0.377838	−	0.925872i	$-0.623332\pi$
$683$	−19.7490	−0.755675	−0.377838	+	0.925872i	$0.623332\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	8.70850	0.332250
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	43.0405	1.63734	0.818669	−	0.574265i	$-0.194712\pi$
$691$	43.0405	1.63734	0.818669	+	0.574265i	$0.194712\pi$
$692$	0	0
$693$	0.583005	0.0221465
$694$	0	0
$695$	−17.8745	−0.678019
$696$	0	0
$697$	38.5830	1.46144
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	16.5830	0.626331	0.313166	−	0.949698i	$-0.398610\pi$
$701$	16.5830	0.626331	0.313166	+	0.949698i	$0.398610\pi$
$702$	0	0
$703$	3.64575	0.137502
$704$	0	0
$705$	1.29150	0.0486408
$706$	0	0
$707$	−15.2915	−0.575096
$708$	0	0
$709$	−4.70850	−0.176831	−0.0884157	−	0.996084i	$-0.528180\pi$
$709$	−4.70850	−0.176831	−0.0884157	+	0.996084i	$0.528180\pi$
$710$	0	0
$711$	−6.58301	−0.246882
$712$	0	0
$713$	55.7490	2.08782
$714$	0	0
$715$	0.125492	0.00469314
$716$	0	0
$717$	−15.6458	−0.584301
$718$	0	0
$719$	−32.8118	−1.22367	−0.611836	−	0.790985i	$-0.709569\pi$
$719$	−32.8118	−1.22367	−0.611836	+	0.790985i	$0.709569\pi$
$720$	0	0
$721$	−17.4170	−0.648643
$722$	0	0
$723$	16.5830	0.616729
$724$	0	0
$725$	8.93725	0.331921
$726$	0	0
$727$	−14.1033	−0.523061	−0.261531	−	0.965195i	$-0.584227\pi$
$727$	−14.1033	−0.523061	−0.261531	+	0.965195i	$0.584227\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.5830	0.835263
$732$	0	0
$733$	7.41699	0.273953	0.136976	−	0.990574i	$-0.456262\pi$
$733$	7.41699	0.273953	0.136976	+	0.990574i	$0.456262\pi$
$734$	0	0
$735$	−4.29150	−0.158294
$736$	0	0
$737$	−2.33202	−0.0859011
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−0.354249	−0.0130137
$742$	0	0
$743$	10.5830	0.388253	0.194126	−	0.980977i	$-0.437813\pi$
$743$	10.5830	0.388253	0.194126	+	0.980977i	$0.437813\pi$
$744$	0	0
$745$	−20.5830	−0.754103
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	44.5830	1.62686	0.813428	−	0.581665i	$-0.197599\pi$
$751$	44.5830	1.62686	0.813428	+	0.581665i	$0.197599\pi$
$752$	0	0
$753$	16.3542	0.595982
$754$	0	0
$755$	8.58301	0.312368
$756$	0	0
$757$	−23.8745	−0.867734	−0.433867	−	0.900977i	$-0.642851\pi$
$757$	−23.8745	−0.867734	−0.433867	+	0.900977i	$0.642851\pi$
$758$	0	0
$759$	3.29150	0.119474
$760$	0	0
$761$	25.7490	0.933401	0.466701	−	0.884415i	$-0.345443\pi$
$761$	25.7490	0.933401	0.466701	+	0.884415i	$0.345443\pi$
$762$	0	0
$763$	−8.70850	−0.315269
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	−17.7490	−0.640046	−0.320023	−	0.947410i	$-0.603691\pi$
$769$	−17.7490	−0.640046	−0.320023	+	0.947410i	$0.603691\pi$
$770$	0	0
$771$	−5.41699	−0.195088
$772$	0	0
$773$	−8.70850	−0.313223	−0.156611	−	0.987660i	$-0.550057\pi$
$773$	−8.70850	−0.313223	−0.156611	+	0.987660i	$0.550057\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	−9.64575	−0.345595
$780$	0	0
$781$	2.58301	0.0924272
$782$	0	0
$783$	8.93725	0.319391
$784$	0	0
$785$	−13.2915	−0.474394
$786$	0	0
$787$	−37.8745	−1.35008	−0.675040	−	0.737781i	$-0.735874\pi$
$787$	−37.8745	−1.35008	−0.675040	+	0.737781i	$0.735874\pi$
$788$	0	0
$789$	−4.58301	−0.163159
$790$	0	0
$791$	6.58301	0.234065
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	11.2915	0.400468
$796$	0	0
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	−5.16601	−0.182760
$800$	0	0
$801$	−16.9373	−0.598448
$802$	0	0
$803$	−3.54249	−0.125012
$804$	0	0
$805$	15.2915	0.538955
$806$	0	0
$807$	4.22876	0.148859
$808$	0	0
$809$	−19.4170	−0.682665	−0.341333	−	0.939943i	$-0.610878\pi$
$809$	−19.4170	−0.682665	−0.341333	+	0.939943i	$0.610878\pi$
$810$	0	0
$811$	−38.3320	−1.34602	−0.673010	−	0.739634i	$-0.734999\pi$
$811$	−38.3320	−1.34602	−0.673010	+	0.739634i	$0.734999\pi$
$812$	0	0
$813$	4.70850	0.165134
$814$	0	0
$815$	2.35425	0.0824657
$816$	0	0
$817$	−5.64575	−0.197520
$818$	0	0
$819$	0.583005	0.0203719
$820$	0	0
$821$	47.6235	1.66207	0.831036	−	0.556218i	$-0.187748\pi$
$821$	47.6235	1.66207	0.831036	+	0.556218i	$0.187748\pi$
$822$	0	0
$823$	−4.22876	−0.147405	−0.0737026	−	0.997280i	$-0.523482\pi$
$823$	−4.22876	−0.147405	−0.0737026	+	0.997280i	$0.523482\pi$
$824$	0	0
$825$	−0.354249	−0.0123334
$826$	0	0
$827$	−30.4575	−1.05911	−0.529556	−	0.848275i	$-0.677641\pi$
$827$	−30.4575	−1.05911	−0.529556	+	0.848275i	$0.677641\pi$
$828$	0	0
$829$	2.70850	0.0940700	0.0470350	−	0.998893i	$-0.485023\pi$
$829$	2.70850	0.0940700	0.0470350	+	0.998893i	$0.485023\pi$
$830$	0	0
$831$	0.583005	0.0202242
$832$	0	0
$833$	17.1660	0.594767
$834$	0	0
$835$	21.2915	0.736823
$836$	0	0
$837$	−6.00000	−0.207390
$838$	0	0
$839$	12.4575	0.430081	0.215041	−	0.976605i	$-0.431012\pi$
$839$	12.4575	0.430081	0.215041	+	0.976605i	$0.431012\pi$
$840$	0	0
$841$	50.8745	1.75429
$842$	0	0
$843$	−32.2288	−1.11002
$844$	0	0
$845$	−12.8745	−0.442897
$846$	0	0
$847$	17.8967	0.614939
$848$	0	0
$849$	11.5203	0.395374
$850$	0	0
$851$	−33.8745	−1.16120
$852$	0	0
$853$	14.7085	0.503609	0.251805	−	0.967778i	$-0.418976\pi$
$853$	14.7085	0.503609	0.251805	+	0.967778i	$0.418976\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	31.0405	1.06032	0.530162	−	0.847896i	$-0.322131\pi$
$857$	31.0405	1.06032	0.530162	+	0.847896i	$0.322131\pi$
$858$	0	0
$859$	34.3320	1.17139	0.585697	−	0.810530i	$-0.300821\pi$
$859$	34.3320	1.17139	0.585697	+	0.810530i	$0.300821\pi$
$860$	0	0
$861$	15.8745	0.541002
$862$	0	0
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	2.33202	0.0791084
$870$	0	0
$871$	−2.33202	−0.0790175
$872$	0	0
$873$	−2.93725	−0.0994110
$874$	0	0
$875$	−1.64575	−0.0556365
$876$	0	0
$877$	−25.0627	−0.846309	−0.423154	−	0.906058i	$-0.639077\pi$
$877$	−25.0627	−0.846309	−0.423154	+	0.906058i	$0.639077\pi$
$878$	0	0
$879$	5.41699	0.182711
$880$	0	0
$881$	−0.125492	−0.00422794	−0.00211397	−	0.999998i	$-0.500673\pi$
$881$	−0.125492	−0.00422794	−0.00211397	+	0.999998i	$0.500673\pi$
$882$	0	0
$883$	−10.8118	−0.363845	−0.181922	−	0.983313i	$-0.558232\pi$
$883$	−10.8118	−0.363845	−0.181922	+	0.983313i	$0.558232\pi$
$884$	0	0
$885$	−11.2915	−0.379560
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	6.58301	0.220787
$890$	0	0
$891$	−0.354249	−0.0118678
$892$	0	0
$893$	1.29150	0.0432185
$894$	0	0
$895$	7.29150	0.243728
$896$	0	0
$897$	3.29150	0.109900
$898$	0	0
$899$	−53.6235	−1.78844
$900$	0	0
$901$	−45.1660	−1.50470
$902$	0	0
$903$	9.29150	0.309202
$904$	0	0
$905$	17.2915	0.574789
$906$	0	0
$907$	47.0405	1.56195	0.780977	−	0.624559i	$-0.214721\pi$
$907$	47.0405	1.56195	0.780977	+	0.624559i	$0.214721\pi$
$908$	0	0
$909$	9.29150	0.308180
$910$	0	0
$911$	−26.5830	−0.880734	−0.440367	−	0.897818i	$-0.645152\pi$
$911$	−26.5830	−0.880734	−0.440367	+	0.897818i	$0.645152\pi$
$912$	0	0
$913$	2.12549	0.0703435
$914$	0	0
$915$	−11.2915	−0.373286
$916$	0	0
$917$	−23.4170	−0.773297
$918$	0	0
$919$	−14.8340	−0.489328	−0.244664	−	0.969608i	$-0.578678\pi$
$919$	−14.8340	−0.489328	−0.244664	+	0.969608i	$0.578678\pi$
$920$	0	0
$921$	24.4575	0.805902
$922$	0	0
$923$	2.58301	0.0850207
$924$	0	0
$925$	3.64575	0.119872
$926$	0	0
$927$	10.5830	0.347591
$928$	0	0
$929$	−11.8745	−0.389590	−0.194795	−	0.980844i	$-0.562404\pi$
$929$	−11.8745	−0.389590	−0.194795	+	0.980844i	$0.562404\pi$
$930$	0	0
$931$	−4.29150	−0.140648
$932$	0	0
$933$	−22.9373	−0.750932
$934$	0	0
$935$	1.41699	0.0463407
$936$	0	0
$937$	20.1255	0.657471	0.328736	−	0.944422i	$-0.393378\pi$
$937$	20.1255	0.657471	0.328736	+	0.944422i	$0.393378\pi$
$938$	0	0
$939$	17.2915	0.564287
$940$	0	0
$941$	−11.7712	−0.383732	−0.191866	−	0.981421i	$-0.561454\pi$
$941$	−11.7712	−0.383732	−0.191866	+	0.981421i	$0.561454\pi$
$942$	0	0
$943$	89.6235	2.91854
$944$	0	0
$945$	−1.64575	−0.0535363
$946$	0	0
$947$	−11.4170	−0.371002	−0.185501	−	0.982644i	$-0.559391\pi$
$947$	−11.4170	−0.371002	−0.185501	+	0.982644i	$0.559391\pi$
$948$	0	0
$949$	−3.54249	−0.114994
$950$	0	0
$951$	20.4575	0.663380
$952$	0	0
$953$	−10.5830	−0.342817	−0.171409	−	0.985200i	$-0.554832\pi$
$953$	−10.5830	−0.342817	−0.171409	+	0.985200i	$0.554832\pi$
$954$	0	0
$955$	−6.22876	−0.201558
$956$	0	0
$957$	−3.16601	−0.102343
$958$	0	0
$959$	−9.87451	−0.318864
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−11.6458	−0.374890
$966$	0	0
$967$	41.6458	1.33924	0.669619	−	0.742705i	$-0.266458\pi$
$967$	41.6458	1.33924	0.669619	+	0.742705i	$0.266458\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	29.4170	0.943066
$974$	0	0
$975$	−0.354249	−0.0113450
$976$	0	0
$977$	−51.0405	−1.63293	−0.816465	−	0.577394i	$-0.804070\pi$
$977$	−51.0405	−1.63293	−0.816465	+	0.577394i	$0.804070\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	5.29150	0.168945
$982$	0	0
$983$	−34.4575	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$983$	−34.4575	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$984$	0	0
$985$	−25.1660	−0.801856
$986$	0	0
$987$	−2.12549	−0.0676552
$988$	0	0
$989$	52.4575	1.66805
$990$	0	0
$991$	35.7490	1.13560	0.567802	−	0.823165i	$-0.307794\pi$
$991$	35.7490	1.13560	0.567802	+	0.823165i	$0.307794\pi$
$992$	0	0
$993$	−11.4170	−0.362307
$994$	0	0
$995$	−7.29150	−0.231156
$996$	0	0
$997$	26.7085	0.845867	0.422933	−	0.906161i	$-0.361000\pi$
$997$	26.7085	0.845867	0.422933	+	0.906161i	$0.361000\pi$
$998$	0	0
$999$	3.64575	0.115346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bo.1.1		2
4.3	odd	2		285.2.a.d.1.1	✓	2
12.11	even	2		855.2.a.g.1.2		2
20.3	even	4		1425.2.c.i.799.3		4
20.7	even	4		1425.2.c.i.799.2		4
20.19	odd	2		1425.2.a.p.1.2		2
60.59	even	2		4275.2.a.u.1.1		2
76.75	even	2		5415.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.d.1.1	✓	2	4.3	odd	2
855.2.a.g.1.2		2	12.11	even	2
1425.2.a.p.1.2		2	20.19	odd	2
1425.2.c.i.799.2		4	20.7	even	4
1425.2.c.i.799.3		4	20.3	even	4
4275.2.a.u.1.1		2	60.59	even	2
4560.2.a.bo.1.1		2	1.1	even	1	trivial
5415.2.a.s.1.2		2	76.75	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.64575 q^{7} +1.00000 q^{9} -0.354249 q^{11} -0.354249 q^{13} +1.00000 q^{15} -4.00000 q^{17} +1.00000 q^{19} -1.64575 q^{21} -9.29150 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.93725 q^{29} -6.00000 q^{31} -0.354249 q^{33} -1.64575 q^{35} +3.64575 q^{37} -0.354249 q^{39} -9.64575 q^{41} -5.64575 q^{43} +1.00000 q^{45} +1.29150 q^{47} -4.29150 q^{49} -4.00000 q^{51} +11.2915 q^{53} -0.354249 q^{55} +1.00000 q^{57} -11.2915 q^{59} -11.2915 q^{61} -1.64575 q^{63} -0.354249 q^{65} +6.58301 q^{67} -9.29150 q^{69} -7.29150 q^{71} +10.0000 q^{73} +1.00000 q^{75} +0.583005 q^{77} -6.58301 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} +8.93725 q^{87} -16.9373 q^{89} +0.583005 q^{91} -6.00000 q^{93} +1.00000 q^{95} -2.93725 q^{97} -0.354249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^17 + 2 * q^19 + 2 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 - 12 * q^31 - 6 * q^33 + 2 * q^35 + 2 * q^37 - 6 * q^39 - 14 * q^41 - 6 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 8 * q^51 + 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 12 * q^61 + 2 * q^63 - 6 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 20 * q^73 + 2 * q^75 - 20 * q^77 + 8 * q^79 + 2 * q^81 - 12 * q^83 - 8 * q^85 + 2 * q^87 - 18 * q^89 - 20 * q^91 - 12 * q^93 + 2 * q^95 + 10 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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