LMFDB - Embedded newform 1856.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.a (trivial)

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:	yes
Analytic conductor:	$14.8202346151$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{3} -3.00000 q^{5} +2.00000 q^{9} +O(q^{10})$	$q-2.23607 q^{3} -3.00000 q^{5} +2.00000 q^{9} +2.23607 q^{11} -1.00000 q^{13} +6.70820 q^{15} +2.00000 q^{17} +4.47214 q^{23} +4.00000 q^{25} +2.23607 q^{27} +1.00000 q^{29} +2.23607 q^{31} -5.00000 q^{33} -8.00000 q^{37} +2.23607 q^{39} +11.1803 q^{43} -6.00000 q^{45} -6.70820 q^{47} -7.00000 q^{49} -4.47214 q^{51} +1.00000 q^{53} -6.70820 q^{55} +4.47214 q^{59} -10.0000 q^{61} +3.00000 q^{65} +8.94427 q^{67} -10.0000 q^{69} -13.4164 q^{71} +4.00000 q^{73} -8.94427 q^{75} +6.70820 q^{79} -11.0000 q^{81} -13.4164 q^{83} -6.00000 q^{85} -2.23607 q^{87} -5.00000 q^{93} -12.0000 q^{97} +4.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 4 q^{9}+O(q^{10}) $ 2 * q - 6 * q^5 + 4 * q^9	$ 2 q - 6 q^{5} + 4 q^{9} - 2 q^{13} + 4 q^{17} + 8 q^{25} + 2 q^{29} - 10 q^{33} - 16 q^{37} - 12 q^{45} - 14 q^{49} + 2 q^{53} - 20 q^{61} + 6 q^{65} - 20 q^{69} + 8 q^{73} - 22 q^{81} - 12 q^{85} - 10 q^{93} - 24 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 4 * q^9 - 2 * q^13 + 4 * q^17 + 8 * q^25 + 2 * q^29 - 10 * q^33 - 16 * q^37 - 12 * q^45 - 14 * q^49 + 2 * q^53 - 20 * q^61 + 6 * q^65 - 20 * q^69 + 8 * q^73 - 22 * q^81 - 12 * q^85 - 10 * q^93 - 24 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	6.70820	1.73205
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	2.23607	0.401610	0.200805	−	0.979631i	$-0.435644\pi$
$31$	2.23607	0.401610	0.200805	+	0.979631i	$0.435644\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	2.23607	0.358057
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	11.1803	1.70499	0.852493	−	0.522739i	$-0.175090\pi$
$43$	11.1803	1.70499	0.852493	+	0.522739i	$0.175090\pi$
$44$	0	0
$45$	−6.00000	−0.894427
$46$	0	0
$47$	−6.70820	−0.978492	−0.489246	−	0.872146i	$-0.662728\pi$
$47$	−6.70820	−0.978492	−0.489246	+	0.872146i	$0.662728\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−6.70820	−0.904534
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	8.94427	1.09272	0.546358	−	0.837552i	$-0.316014\pi$
$67$	8.94427	1.09272	0.546358	+	0.837552i	$0.316014\pi$
$68$	0	0
$69$	−10.0000	−1.20386
$70$	0	0
$71$	−13.4164	−1.59223	−0.796117	−	0.605142i	$-0.793116\pi$
$71$	−13.4164	−1.59223	−0.796117	+	0.605142i	$0.793116\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	−8.94427	−1.03280
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.70820	0.754732	0.377366	−	0.926064i	$-0.376830\pi$
$79$	6.70820	0.754732	0.377366	+	0.926064i	$0.376830\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−13.4164	−1.47264	−0.736321	−	0.676632i	$-0.763439\pi$
$83$	−13.4164	−1.47264	−0.736321	+	0.676632i	$0.763439\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	−2.23607	−0.239732
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.00000	−0.518476
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	4.47214	0.449467
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−13.4164	−1.32196	−0.660979	−	0.750404i	$-0.729859\pi$
$103$	−13.4164	−1.32196	−0.660979	+	0.750404i	$0.729859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	17.8885	1.69791
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−13.4164	−1.25109
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−8.94427	−0.793676	−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427	−0.793676	−0.396838	+	0.917889i	$0.629892\pi$
$128$	0	0
$129$	−25.0000	−2.20113
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.70820	−0.577350
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	−22.3607	−1.89661	−0.948304	−	0.317363i	$-0.897203\pi$
$139$	−22.3607	−1.89661	−0.948304	+	0.317363i	$0.897203\pi$
$140$	0	0
$141$	15.0000	1.26323
$142$	0	0
$143$	−2.23607	−0.186989
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	15.6525	1.29099
$148$	0	0
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	0	0
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−6.70820	−0.538816
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	−2.23607	−0.177332
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.70820	−0.525427	−0.262714	−	0.964874i	$-0.584617\pi$
$163$	−6.70820	−0.525427	−0.262714	+	0.964874i	$0.584617\pi$
$164$	0	0
$165$	15.0000	1.16775
$166$	0	0
$167$	4.47214	0.346064	0.173032	−	0.984916i	$-0.444644\pi$
$167$	4.47214	0.346064	0.173032	+	0.984916i	$0.444644\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	−22.3607	−1.67132	−0.835658	−	0.549250i	$-0.814913\pi$
$179$	−22.3607	−1.67132	−0.835658	+	0.549250i	$0.814913\pi$
$180$	0	0
$181$	23.0000	1.70958	0.854788	−	0.518977i	$-0.173687\pi$
$181$	23.0000	1.70958	0.854788	+	0.518977i	$0.173687\pi$
$182$	0	0
$183$	22.3607	1.65295
$184$	0	0
$185$	24.0000	1.76452
$186$	0	0
$187$	4.47214	0.327035
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.94427	0.647185	0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	−6.70820	−0.480384
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−8.94427	−0.634043	−0.317021	−	0.948418i	$-0.602683\pi$
$199$	−8.94427	−0.634043	−0.317021	+	0.948418i	$0.602683\pi$
$200$	0	0
$201$	−20.0000	−1.41069
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.94427	0.621670
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−15.6525	−1.07756	−0.538780	−	0.842446i	$-0.681115\pi$
$211$	−15.6525	−1.07756	−0.538780	+	0.842446i	$0.681115\pi$
$212$	0	0
$213$	30.0000	2.05557
$214$	0	0
$215$	−33.5410	−2.28748
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.94427	−0.604398
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	−4.47214	−0.299476	−0.149738	−	0.988726i	$-0.547843\pi$
$223$	−4.47214	−0.299476	−0.149738	+	0.988726i	$0.547843\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	13.4164	0.890478	0.445239	−	0.895412i	$-0.353119\pi$
$227$	13.4164	0.890478	0.445239	+	0.895412i	$0.353119\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	20.1246	1.31278
$236$	0	0
$237$	−15.0000	−0.974355
$238$	0	0
$239$	−22.3607	−1.44639	−0.723196	−	0.690643i	$-0.757328\pi$
$239$	−22.3607	−1.44639	−0.723196	+	0.690643i	$0.757328\pi$
$240$	0	0
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	0	0
$243$	17.8885	1.14755
$244$	0	0
$245$	21.0000	1.34164
$246$	0	0
$247$	0	0
$248$	0	0
$249$	30.0000	1.90117
$250$	0	0
$251$	24.5967	1.55253	0.776266	−	0.630405i	$-0.217111\pi$
$251$	24.5967	1.55253	0.776266	+	0.630405i	$0.217111\pi$
$252$	0	0
$253$	10.0000	0.628695
$254$	0	0
$255$	13.4164	0.840168
$256$	0	0
$257$	17.0000	1.06043	0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000	1.06043	0.530215	+	0.847863i	$0.322111\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	29.0689	1.79246	0.896232	−	0.443585i	$-0.146294\pi$
$263$	29.0689	1.79246	0.896232	+	0.443585i	$0.146294\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−11.1803	−0.679157	−0.339579	−	0.940578i	$-0.610284\pi$
$271$	−11.1803	−0.679157	−0.339579	+	0.940578i	$0.610284\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.94427	0.539360
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	4.47214	0.267740
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	22.3607	1.32920	0.664602	−	0.747197i	$-0.268601\pi$
$283$	22.3607	1.32920	0.664602	+	0.747197i	$0.268601\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	26.8328	1.57297
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−13.4164	−0.781133
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	−4.47214	−0.258630
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.8885	1.02767
$304$	0	0
$305$	30.0000	1.71780
$306$	0	0
$307$	−33.5410	−1.91429	−0.957144	−	0.289614i	$-0.906473\pi$
$307$	−33.5410	−1.91429	−0.957144	+	0.289614i	$0.906473\pi$
$308$	0	0
$309$	30.0000	1.70664
$310$	0	0
$311$	8.94427	0.507183	0.253592	−	0.967311i	$-0.418388\pi$
$311$	8.94427	0.507183	0.253592	+	0.967311i	$0.418388\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	2.23607	0.125196
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	24.5967	1.36020
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.23607	0.122905	0.0614527	−	0.998110i	$-0.480427\pi$
$331$	2.23607	0.122905	0.0614527	+	0.998110i	$0.480427\pi$
$332$	0	0
$333$	−16.0000	−0.876795
$334$	0	0
$335$	−26.8328	−1.46603
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−13.4164	−0.728679
$340$	0	0
$341$	5.00000	0.270765
$342$	0	0
$343$	0	0
$344$	0	0
$345$	30.0000	1.61515
$346$	0	0
$347$	4.47214	0.240077	0.120038	−	0.992769i	$-0.461698\pi$
$347$	4.47214	0.240077	0.120038	+	0.992769i	$0.461698\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	−2.23607	−0.119352
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	40.2492	2.13621
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.6525	0.826106	0.413053	−	0.910707i	$-0.364462\pi$
$359$	15.6525	0.826106	0.413053	+	0.910707i	$0.364462\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	13.4164	0.704179
$364$	0	0
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−35.7771	−1.86755	−0.933774	−	0.357862i	$-0.883506\pi$
$367$	−35.7771	−1.86755	−0.933774	+	0.357862i	$0.883506\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	−6.70820	−0.346410
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−8.94427	−0.459436	−0.229718	−	0.973257i	$-0.573780\pi$
$379$	−8.94427	−0.459436	−0.229718	+	0.973257i	$0.573780\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.3607	1.13666
$388$	0	0
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	8.94427	0.452331
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−20.1246	−1.01258
$396$	0	0
$397$	27.0000	1.35509	0.677546	−	0.735481i	$-0.263044\pi$
$397$	27.0000	1.35509	0.677546	+	0.735481i	$0.263044\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	−2.23607	−0.111386
$404$	0	0
$405$	33.0000	1.63978
$406$	0	0
$407$	−17.8885	−0.886702
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−17.8885	−0.882377
$412$	0	0
$413$	0	0
$414$	0	0
$415$	40.2492	1.97576
$416$	0	0
$417$	50.0000	2.44851
$418$	0	0
$419$	−22.3607	−1.09239	−0.546195	−	0.837658i	$-0.683924\pi$
$419$	−22.3607	−1.09239	−0.546195	+	0.837658i	$0.683924\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	−13.4164	−0.652328
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.00000	0.241402
$430$	0	0
$431$	22.3607	1.07708	0.538538	−	0.842601i	$-0.318977\pi$
$431$	22.3607	1.07708	0.538538	+	0.842601i	$0.318977\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	6.70820	0.321634
$436$	0	0
$437$	0	0
$438$	0	0
$439$	13.4164	0.640330	0.320165	−	0.947362i	$-0.396262\pi$
$439$	13.4164	0.640330	0.320165	+	0.947362i	$0.396262\pi$
$440$	0	0
$441$	−14.0000	−0.666667
$442$	0	0
$443$	−35.7771	−1.69982	−0.849910	−	0.526927i	$-0.823344\pi$
$443$	−35.7771	−1.69982	−0.849910	+	0.526927i	$0.823344\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−24.5967	−1.16339
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	4.47214	0.208741
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	40.2492	1.87054	0.935270	−	0.353935i	$-0.115157\pi$
$463$	40.2492	1.87054	0.935270	+	0.353935i	$0.115157\pi$
$464$	0	0
$465$	15.0000	0.695608
$466$	0	0
$467$	11.1803	0.517364	0.258682	−	0.965962i	$-0.416712\pi$
$467$	11.1803	0.517364	0.258682	+	0.965962i	$0.416712\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−17.8885	−0.824261
$472$	0	0
$473$	25.0000	1.14950
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−2.23607	−0.102169	−0.0510843	−	0.998694i	$-0.516268\pi$
$479$	−2.23607	−0.102169	−0.0510843	+	0.998694i	$0.516268\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.0000	1.63468
$486$	0	0
$487$	8.94427	0.405304	0.202652	−	0.979251i	$-0.435044\pi$
$487$	8.94427	0.405304	0.202652	+	0.979251i	$0.435044\pi$
$488$	0	0
$489$	15.0000	0.678323
$490$	0	0
$491$	−38.0132	−1.71551	−0.857755	−	0.514059i	$-0.828141\pi$
$491$	−38.0132	−1.71551	−0.857755	+	0.514059i	$0.828141\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−13.4164	−0.603023
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−31.3050	−1.40140	−0.700701	−	0.713455i	$-0.747129\pi$
$499$	−31.3050	−1.40140	−0.700701	+	0.713455i	$0.747129\pi$
$500$	0	0
$501$	−10.0000	−0.446767
$502$	0	0
$503$	15.6525	0.697909	0.348955	−	0.937140i	$-0.386537\pi$
$503$	15.6525	0.697909	0.348955	+	0.937140i	$0.386537\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	26.8328	1.19169
$508$	0	0
$509$	−35.0000	−1.55135	−0.775674	−	0.631134i	$-0.782590\pi$
$509$	−35.0000	−1.55135	−0.775674	+	0.631134i	$0.782590\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40.2492	1.77359
$516$	0	0
$517$	−15.0000	−0.659699
$518$	0	0
$519$	31.3050	1.37413
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	−26.8328	−1.17332	−0.586659	−	0.809834i	$-0.699557\pi$
$523$	−26.8328	−1.17332	−0.586659	+	0.809834i	$0.699557\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.47214	0.194809
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	8.94427	0.388148
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−26.8328	−1.16008
$536$	0	0
$537$	50.0000	2.15766
$538$	0	0
$539$	−15.6525	−0.674200
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	0	0
$543$	−51.4296	−2.20705
$544$	0	0
$545$	33.0000	1.41356
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	−20.0000	−0.853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−53.6656	−2.27798
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−11.1803	−0.472878
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	11.1803	0.471195	0.235598	−	0.971851i	$-0.424295\pi$
$563$	11.1803	0.471195	0.235598	+	0.971851i	$0.424295\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	8.94427	0.374306	0.187153	−	0.982331i	$-0.440074\pi$
$571$	8.94427	0.374306	0.187153	+	0.982331i	$0.440074\pi$
$572$	0	0
$573$	−20.0000	−0.835512
$574$	0	0
$575$	17.8885	0.746004
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	8.94427	0.371711
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.23607	0.0926085
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	−4.47214	−0.184585	−0.0922924	−	0.995732i	$-0.529419\pi$
$587$	−4.47214	−0.184585	−0.0922924	+	0.995732i	$0.529419\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−49.1935	−2.02355
$592$	0	0
$593$	−41.0000	−1.68367	−0.841834	−	0.539736i	$-0.818524\pi$
$593$	−41.0000	−1.68367	−0.841834	+	0.539736i	$0.818524\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000	0.818546
$598$	0	0
$599$	−29.0689	−1.18772	−0.593861	−	0.804568i	$-0.702397\pi$
$599$	−29.0689	−1.18772	−0.593861	+	0.804568i	$0.702397\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	17.8885	0.728478
$604$	0	0
$605$	18.0000	0.731804
$606$	0	0
$607$	−20.1246	−0.816833	−0.408416	−	0.912796i	$-0.633919\pi$
$607$	−20.1246	−0.816833	−0.408416	+	0.912796i	$0.633919\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.70820	0.271385
$612$	0	0
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	−42.4853	−1.70763	−0.853814	−	0.520578i	$-0.825716\pi$
$619$	−42.4853	−1.70763	−0.853814	+	0.520578i	$0.825716\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	4.47214	0.178033	0.0890165	−	0.996030i	$-0.471628\pi$
$631$	4.47214	0.178033	0.0890165	+	0.996030i	$0.471628\pi$
$632$	0	0
$633$	35.0000	1.39113
$634$	0	0
$635$	26.8328	1.06483
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	−26.8328	−1.06149
$640$	0	0
$641$	−50.0000	−1.97488	−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−50.0000	−1.97488	−0.987441	+	0.157991i	$0.949498\pi$
$642$	0	0
$643$	−26.8328	−1.05818	−0.529091	−	0.848565i	$-0.677467\pi$
$643$	−26.8328	−1.05818	−0.529091	+	0.848565i	$0.677467\pi$
$644$	0	0
$645$	75.0000	2.95312
$646$	0	0
$647$	−13.4164	−0.527453	−0.263727	−	0.964597i	$-0.584952\pi$
$647$	−13.4164	−0.527453	−0.263727	+	0.964597i	$0.584952\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	−29.0689	−1.13236	−0.566181	−	0.824281i	$-0.691580\pi$
$659$	−29.0689	−1.13236	−0.566181	+	0.824281i	$0.691580\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	4.47214	0.173683
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.47214	0.173162
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	−22.3607	−0.863224
$672$	0	0
$673$	9.00000	0.346925	0.173462	−	0.984841i	$-0.444505\pi$
$673$	9.00000	0.346925	0.173462	+	0.984841i	$0.444505\pi$
$674$	0	0
$675$	8.94427	0.344265
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−30.0000	−1.14960
$682$	0	0
$683$	44.7214	1.71122	0.855608	−	0.517625i	$-0.173184\pi$
$683$	44.7214	1.71122	0.855608	+	0.517625i	$0.173184\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	17.8885	0.680512	0.340256	−	0.940333i	$-0.389486\pi$
$691$	17.8885	0.680512	0.340256	+	0.940333i	$0.389486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	67.0820	2.54457
$696$	0	0
$697$	0	0
$698$	0	0
$699$	20.1246	0.761183
$700$	0	0
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−45.0000	−1.69480
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	13.4164	0.503155
$712$	0	0
$713$	10.0000	0.374503
$714$	0	0
$715$	6.70820	0.250873
$716$	0	0
$717$	50.0000	1.86728
$718$	0	0
$719$	4.47214	0.166783	0.0833913	−	0.996517i	$-0.473425\pi$
$719$	4.47214	0.166783	0.0833913	+	0.996517i	$0.473425\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	33.5410	1.24740
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	26.8328	0.995174	0.497587	−	0.867414i	$-0.334220\pi$
$727$	26.8328	0.995174	0.497587	+	0.867414i	$0.334220\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	22.3607	0.827040
$732$	0	0
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	0	0
$735$	−46.9574	−1.73205
$736$	0	0
$737$	20.0000	0.736709
$738$	0	0
$739$	24.5967	0.904806	0.452403	−	0.891814i	$-0.350567\pi$
$739$	24.5967	0.904806	0.452403	+	0.891814i	$0.350567\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.94427	−0.328134	−0.164067	−	0.986449i	$-0.552461\pi$
$743$	−8.94427	−0.328134	−0.164067	+	0.986449i	$0.552461\pi$
$744$	0	0
$745$	−33.0000	−1.20903
$746$	0	0
$747$	−26.8328	−0.981761
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	−55.0000	−2.00431
$754$	0	0
$755$	−26.8328	−0.976546
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−22.3607	−0.811641
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−12.0000	−0.433861
$766$	0	0
$767$	−4.47214	−0.161479
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−38.0132	−1.36901
$772$	0	0
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	8.94427	0.321288
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−30.0000	−1.07348
$782$	0	0
$783$	2.23607	0.0799106
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	22.3607	0.797072	0.398536	−	0.917153i	$-0.369518\pi$
$787$	22.3607	0.797072	0.398536	+	0.917153i	$0.369518\pi$
$788$	0	0
$789$	−65.0000	−2.31406
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	6.70820	0.237915
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−13.4164	−0.474638
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.94427	0.315637
$804$	0	0
$805$	0	0
$806$	0	0
$807$	31.3050	1.10199
$808$	0	0
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	0	0
$811$	53.6656	1.88446	0.942228	−	0.334973i	$-0.108727\pi$
$811$	53.6656	1.88446	0.942228	+	0.334973i	$0.108727\pi$
$812$	0	0
$813$	25.0000	0.876788
$814$	0	0
$815$	20.1246	0.704934
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	0	0
$823$	26.8328	0.935333	0.467667	−	0.883905i	$-0.345095\pi$
$823$	26.8328	0.935333	0.467667	+	0.883905i	$0.345095\pi$
$824$	0	0
$825$	−20.0000	−0.696311
$826$	0	0
$827$	−33.5410	−1.16634	−0.583168	−	0.812352i	$-0.698187\pi$
$827$	−33.5410	−1.16634	−0.583168	+	0.812352i	$0.698187\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	49.1935	1.70650
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	−13.4164	−0.464294
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	−42.4853	−1.46676	−0.733378	−	0.679822i	$-0.762057\pi$
$839$	−42.4853	−1.46676	−0.733378	+	0.679822i	$0.762057\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−33.5410	−1.15521
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−50.0000	−1.71600
$850$	0	0
$851$	−35.7771	−1.22642
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.0000	0.444072	0.222036	−	0.975039i	$-0.428730\pi$
$857$	13.0000	0.444072	0.222036	+	0.975039i	$0.428730\pi$
$858$	0	0
$859$	−2.23607	−0.0762937	−0.0381468	−	0.999272i	$-0.512145\pi$
$859$	−2.23607	−0.0762937	−0.0381468	+	0.999272i	$0.512145\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.3050	1.06563	0.532816	−	0.846231i	$-0.321134\pi$
$863$	31.3050	1.06563	0.532816	+	0.846231i	$0.321134\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	0	0
$867$	29.0689	0.987231
$868$	0	0
$869$	15.0000	0.508840
$870$	0	0
$871$	−8.94427	−0.303065
$872$	0	0
$873$	−24.0000	−0.812277
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.0000	1.11433	0.557165	−	0.830402i	$-0.311889\pi$
$877$	33.0000	1.11433	0.557165	+	0.830402i	$0.311889\pi$
$878$	0	0
$879$	13.4164	0.452524
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	26.8328	0.902996	0.451498	−	0.892272i	$-0.350890\pi$
$883$	26.8328	0.902996	0.451498	+	0.892272i	$0.350890\pi$
$884$	0	0
$885$	30.0000	1.00844
$886$	0	0
$887$	−33.5410	−1.12620	−0.563099	−	0.826390i	$-0.690391\pi$
$887$	−33.5410	−1.12620	−0.563099	+	0.826390i	$0.690391\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−24.5967	−0.824022
$892$	0	0
$893$	0	0
$894$	0	0
$895$	67.0820	2.24231
$896$	0	0
$897$	10.0000	0.333890
$898$	0	0
$899$	2.23607	0.0745770
$900$	0	0
$901$	2.00000	0.0666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−69.0000	−2.29364
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−16.0000	−0.530687
$910$	0	0
$911$	24.5967	0.814927	0.407463	−	0.913222i	$-0.366413\pi$
$911$	24.5967	0.814927	0.407463	+	0.913222i	$0.366413\pi$
$912$	0	0
$913$	−30.0000	−0.992855
$914$	0	0
$915$	−67.0820	−2.21766
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−13.4164	−0.442566	−0.221283	−	0.975210i	$-0.571025\pi$
$919$	−13.4164	−0.442566	−0.221283	+	0.975210i	$0.571025\pi$
$920$	0	0
$921$	75.0000	2.47133
$922$	0	0
$923$	13.4164	0.441606
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	−26.8328	−0.881305
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−20.0000	−0.654771
$934$	0	0
$935$	−13.4164	−0.438763
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	42.4853	1.38646
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.4853	−1.38059	−0.690293	−	0.723530i	$-0.742518\pi$
$947$	−42.4853	−1.38059	−0.690293	+	0.723530i	$0.742518\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	49.1935	1.59521
$952$	0	0
$953$	11.0000	0.356325	0.178162	−	0.984001i	$-0.442985\pi$
$953$	11.0000	0.356325	0.178162	+	0.984001i	$0.442985\pi$
$954$	0	0
$955$	−26.8328	−0.868290
$956$	0	0
$957$	−5.00000	−0.161627
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.0000	−0.838710
$962$	0	0
$963$	17.8885	0.576450
$964$	0	0
$965$	12.0000	0.386294
$966$	0	0
$967$	−6.70820	−0.215721	−0.107861	−	0.994166i	$-0.534400\pi$
$967$	−6.70820	−0.215721	−0.107861	+	0.994166i	$0.534400\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53.6656	1.72221	0.861106	−	0.508425i	$-0.169772\pi$
$971$	53.6656	1.72221	0.861106	+	0.508425i	$0.169772\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8.94427	0.286446
$976$	0	0
$977$	53.0000	1.69562	0.847810	−	0.530300i	$-0.177921\pi$
$977$	53.0000	1.69562	0.847810	+	0.530300i	$0.177921\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	2.23607	0.0713195	0.0356597	−	0.999364i	$-0.488647\pi$
$983$	2.23607	0.0713195	0.0356597	+	0.999364i	$0.488647\pi$
$984$	0	0
$985$	−66.0000	−2.10293
$986$	0	0
$987$	0	0
$988$	0	0
$989$	50.0000	1.58991
$990$	0	0
$991$	31.3050	0.994435	0.497217	−	0.867626i	$-0.334355\pi$
$991$	31.3050	0.994435	0.497217	+	0.867626i	$0.334355\pi$
$992$	0	0
$993$	−5.00000	−0.158670
$994$	0	0
$995$	26.8328	0.850657
$996$	0	0
$997$	−52.0000	−1.64686	−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000	−1.64686	−0.823428	+	0.567420i	$0.807941\pi$
$998$	0	0
$999$	−17.8885	−0.565968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.t.1.1		2
4.3	odd	2	inner	1856.2.a.t.1.2		2
8.3	odd	2		928.2.a.d.1.1	✓	2
8.5	even	2		928.2.a.d.1.2	yes	2
24.5	odd	2		8352.2.a.k.1.2		2
24.11	even	2		8352.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.d.1.1	✓	2	8.3	odd	2
928.2.a.d.1.2	yes	2	8.5	even	2
1856.2.a.t.1.1		2	1.1	even	1	trivial
1856.2.a.t.1.2		2	4.3	odd	2	inner
8352.2.a.k.1.1		2	24.11	even	2
8352.2.a.k.1.2		2	24.5	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{3} -3.00000 q^{5} +2.00000 q^{9} +O(q^{10})\)	\(q-2.23607 q^{3} -3.00000 q^{5} +2.00000 q^{9} +2.23607 q^{11} -1.00000 q^{13} +6.70820 q^{15} +2.00000 q^{17} +4.47214 q^{23} +4.00000 q^{25} +2.23607 q^{27} +1.00000 q^{29} +2.23607 q^{31} -5.00000 q^{33} -8.00000 q^{37} +2.23607 q^{39} +11.1803 q^{43} -6.00000 q^{45} -6.70820 q^{47} -7.00000 q^{49} -4.47214 q^{51} +1.00000 q^{53} -6.70820 q^{55} +4.47214 q^{59} -10.0000 q^{61} +3.00000 q^{65} +8.94427 q^{67} -10.0000 q^{69} -13.4164 q^{71} +4.00000 q^{73} -8.94427 q^{75} +6.70820 q^{79} -11.0000 q^{81} -13.4164 q^{83} -6.00000 q^{85} -2.23607 q^{87} -5.00000 q^{93} -12.0000 q^{97} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 4 q^{9}+O(q^{10}) \) 2 * q - 6 * q^5 + 4 * q^9	\( 2 q - 6 q^{5} + 4 q^{9} - 2 q^{13} + 4 q^{17} + 8 q^{25} + 2 q^{29} - 10 q^{33} - 16 q^{37} - 12 q^{45} - 14 q^{49} + 2 q^{53} - 20 q^{61} + 6 q^{65} - 20 q^{69} + 8 q^{73} - 22 q^{81} - 12 q^{85} - 10 q^{93} - 24 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 4 * q^9 - 2 * q^13 + 4 * q^17 + 8 * q^25 + 2 * q^29 - 10 * q^33 - 16 * q^37 - 12 * q^45 - 14 * q^49 + 2 * q^53 - 20 * q^61 + 6 * q^65 - 20 * q^69 + 8 * q^73 - 22 * q^81 - 12 * q^85 - 10 * q^93 - 24 * q^97

Introduction

L-functions

Modular forms

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Database

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