LMFDB - Embedded newform 1456.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{3} +1.73205 q^{5} +1.00000 q^{7} +4.46410 q^{9} +O(q^{10})$	$q+2.73205 q^{3} +1.73205 q^{5} +1.00000 q^{7} +4.46410 q^{9} +2.73205 q^{11} -1.00000 q^{13} +4.73205 q^{15} +2.73205 q^{17} -1.73205 q^{19} +2.73205 q^{21} -0.464102 q^{23} -2.00000 q^{25} +4.00000 q^{27} -6.46410 q^{29} +0.267949 q^{31} +7.46410 q^{33} +1.73205 q^{35} -3.26795 q^{37} -2.73205 q^{39} -10.3923 q^{41} +12.4641 q^{43} +7.73205 q^{45} +5.73205 q^{47} +1.00000 q^{49} +7.46410 q^{51} -9.92820 q^{53} +4.73205 q^{55} -4.73205 q^{57} -3.46410 q^{59} -2.00000 q^{61} +4.46410 q^{63} -1.73205 q^{65} +10.3923 q^{67} -1.26795 q^{69} -10.1962 q^{71} +9.73205 q^{73} -5.46410 q^{75} +2.73205 q^{77} -3.92820 q^{79} -2.46410 q^{81} +6.26795 q^{83} +4.73205 q^{85} -17.6603 q^{87} +12.2679 q^{89} -1.00000 q^{91} +0.732051 q^{93} -3.00000 q^{95} -16.1244 q^{97} +12.1962 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 6 q^{15} + 2 q^{17} + 2 q^{21} + 6 q^{23} - 4 q^{25} + 8 q^{27} - 6 q^{29} + 4 q^{31} + 8 q^{33} - 10 q^{37} - 2 q^{39} + 18 q^{43} + 12 q^{45} + 8 q^{47} + 2 q^{49} + 8 q^{51} - 6 q^{53} + 6 q^{55} - 6 q^{57} - 4 q^{61} + 2 q^{63} - 6 q^{69} - 10 q^{71} + 16 q^{73} - 4 q^{75} + 2 q^{77} + 6 q^{79} + 2 q^{81} + 16 q^{83} + 6 q^{85} - 18 q^{87} + 28 q^{89} - 2 q^{91} - 2 q^{93} - 6 q^{95} - 8 q^{97} + 14 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 6 * q^15 + 2 * q^17 + 2 * q^21 + 6 * q^23 - 4 * q^25 + 8 * q^27 - 6 * q^29 + 4 * q^31 + 8 * q^33 - 10 * q^37 - 2 * q^39 + 18 * q^43 + 12 * q^45 + 8 * q^47 + 2 * q^49 + 8 * q^51 - 6 * q^53 + 6 * q^55 - 6 * q^57 - 4 * q^61 + 2 * q^63 - 6 * q^69 - 10 * q^71 + 16 * q^73 - 4 * q^75 + 2 * q^77 + 6 * q^79 + 2 * q^81 + 16 * q^83 + 6 * q^85 - 18 * q^87 + 28 * q^89 - 2 * q^91 - 2 * q^93 - 6 * q^95 - 8 * q^97 + 14 * q^99

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
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	2.73205	0.823744	0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205	0.823744	0.411872	+	0.911242i	$0.364875\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	4.73205	1.22181
$16$	0	0
$17$	2.73205	0.662620	0.331310	−	0.943522i	$-0.392509\pi$
$17$	2.73205	0.662620	0.331310	+	0.943522i	$0.392509\pi$
$18$	0	0
$19$	−1.73205	−0.397360	−0.198680	−	0.980064i	$-0.563665\pi$
$19$	−1.73205	−0.397360	−0.198680	+	0.980064i	$0.563665\pi$
$20$	0	0
$21$	2.73205	0.596182
$22$	0	0
$23$	−0.464102	−0.0967719	−0.0483859	−	0.998829i	$-0.515408\pi$
$23$	−0.464102	−0.0967719	−0.0483859	+	0.998829i	$0.515408\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−6.46410	−1.20035	−0.600177	−	0.799867i	$-0.704903\pi$
$29$	−6.46410	−1.20035	−0.600177	+	0.799867i	$0.704903\pi$
$30$	0	0
$31$	0.267949	0.0481251	0.0240625	−	0.999710i	$-0.492340\pi$
$31$	0.267949	0.0481251	0.0240625	+	0.999710i	$0.492340\pi$
$32$	0	0
$33$	7.46410	1.29933
$34$	0	0
$35$	1.73205	0.292770
$36$	0	0
$37$	−3.26795	−0.537248	−0.268624	−	0.963245i	$-0.586569\pi$
$37$	−3.26795	−0.537248	−0.268624	+	0.963245i	$0.586569\pi$
$38$	0	0
$39$	−2.73205	−0.437478
$40$	0	0
$41$	−10.3923	−1.62301	−0.811503	−	0.584349i	$-0.801350\pi$
$41$	−10.3923	−1.62301	−0.811503	+	0.584349i	$0.801350\pi$
$42$	0	0
$43$	12.4641	1.90076	0.950379	−	0.311095i	$-0.100696\pi$
$43$	12.4641	1.90076	0.950379	+	0.311095i	$0.100696\pi$
$44$	0	0
$45$	7.73205	1.15263
$46$	0	0
$47$	5.73205	0.836106	0.418053	−	0.908423i	$-0.362713\pi$
$47$	5.73205	0.836106	0.418053	+	0.908423i	$0.362713\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.46410	1.04518
$52$	0	0
$53$	−9.92820	−1.36374	−0.681872	−	0.731472i	$-0.738834\pi$
$53$	−9.92820	−1.36374	−0.681872	+	0.731472i	$0.738834\pi$
$54$	0	0
$55$	4.73205	0.638070
$56$	0	0
$57$	−4.73205	−0.626775
$58$	0	0
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	4.46410	0.562424
$64$	0	0
$65$	−1.73205	−0.214834
$66$	0	0
$67$	10.3923	1.26962	0.634811	−	0.772667i	$-0.281078\pi$
$67$	10.3923	1.26962	0.634811	+	0.772667i	$0.281078\pi$
$68$	0	0
$69$	−1.26795	−0.152643
$70$	0	0
$71$	−10.1962	−1.21006	−0.605030	−	0.796202i	$-0.706839\pi$
$71$	−10.1962	−1.21006	−0.605030	+	0.796202i	$0.706839\pi$
$72$	0	0
$73$	9.73205	1.13905	0.569525	−	0.821974i	$-0.307127\pi$
$73$	9.73205	1.13905	0.569525	+	0.821974i	$0.307127\pi$
$74$	0	0
$75$	−5.46410	−0.630940
$76$	0	0
$77$	2.73205	0.311346
$78$	0	0
$79$	−3.92820	−0.441957	−0.220979	−	0.975279i	$-0.570925\pi$
$79$	−3.92820	−0.441957	−0.220979	+	0.975279i	$0.570925\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	6.26795	0.687997	0.343998	−	0.938970i	$-0.388219\pi$
$83$	6.26795	0.687997	0.343998	+	0.938970i	$0.388219\pi$
$84$	0	0
$85$	4.73205	0.513263
$86$	0	0
$87$	−17.6603	−1.89338
$88$	0	0
$89$	12.2679	1.30040	0.650200	−	0.759763i	$-0.274685\pi$
$89$	12.2679	1.30040	0.650200	+	0.759763i	$0.274685\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0.732051	0.0759101
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	−16.1244	−1.63718	−0.818590	−	0.574378i	$-0.805244\pi$
$97$	−16.1244	−1.63718	−0.818590	+	0.574378i	$0.805244\pi$
$98$	0	0
$99$	12.1962	1.22576
$100$	0	0
$101$	10.1962	1.01456	0.507278	−	0.861783i	$-0.330652\pi$
$101$	10.1962	1.01456	0.507278	+	0.861783i	$0.330652\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	4.73205	0.461801
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	−3.80385	−0.364343	−0.182171	−	0.983267i	$-0.558313\pi$
$109$	−3.80385	−0.364343	−0.182171	+	0.983267i	$0.558313\pi$
$110$	0	0
$111$	−8.92820	−0.847428
$112$	0	0
$113$	1.53590	0.144485	0.0722426	−	0.997387i	$-0.476984\pi$
$113$	1.53590	0.144485	0.0722426	+	0.997387i	$0.476984\pi$
$114$	0	0
$115$	−0.803848	−0.0749592
$116$	0	0
$117$	−4.46410	−0.412706
$118$	0	0
$119$	2.73205	0.250447
$120$	0	0
$121$	−3.53590	−0.321445
$122$	0	0
$123$	−28.3923	−2.56005
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−6.39230	−0.567225	−0.283613	−	0.958939i	$-0.591533\pi$
$127$	−6.39230	−0.567225	−0.283613	+	0.958939i	$0.591533\pi$
$128$	0	0
$129$	34.0526	2.99816
$130$	0	0
$131$	−6.53590	−0.571044	−0.285522	−	0.958372i	$-0.592167\pi$
$131$	−6.53590	−0.571044	−0.285522	+	0.958372i	$0.592167\pi$
$132$	0	0
$133$	−1.73205	−0.150188
$134$	0	0
$135$	6.92820	0.596285
$136$	0	0
$137$	7.26795	0.620943	0.310471	−	0.950583i	$-0.399513\pi$
$137$	7.26795	0.620943	0.310471	+	0.950583i	$0.399513\pi$
$138$	0	0
$139$	−18.5885	−1.57665	−0.788326	−	0.615258i	$-0.789052\pi$
$139$	−18.5885	−1.57665	−0.788326	+	0.615258i	$0.789052\pi$
$140$	0	0
$141$	15.6603	1.31883
$142$	0	0
$143$	−2.73205	−0.228466
$144$	0	0
$145$	−11.1962	−0.929790
$146$	0	0
$147$	2.73205	0.225336
$148$	0	0
$149$	17.6603	1.44678	0.723392	−	0.690437i	$-0.242582\pi$
$149$	17.6603	1.44678	0.723392	+	0.690437i	$0.242582\pi$
$150$	0	0
$151$	7.66025	0.623383	0.311691	−	0.950183i	$-0.399105\pi$
$151$	7.66025	0.623383	0.311691	+	0.950183i	$0.399105\pi$
$152$	0	0
$153$	12.1962	0.986000
$154$	0	0
$155$	0.464102	0.0372775
$156$	0	0
$157$	−13.6603	−1.09021	−0.545103	−	0.838369i	$-0.683510\pi$
$157$	−13.6603	−1.09021	−0.545103	+	0.838369i	$0.683510\pi$
$158$	0	0
$159$	−27.1244	−2.15110
$160$	0	0
$161$	−0.464102	−0.0365763
$162$	0	0
$163$	9.46410	0.741286	0.370643	−	0.928775i	$-0.379137\pi$
$163$	9.46410	0.741286	0.370643	+	0.928775i	$0.379137\pi$
$164$	0	0
$165$	12.9282	1.00646
$166$	0	0
$167$	5.19615	0.402090	0.201045	−	0.979582i	$-0.435566\pi$
$167$	5.19615	0.402090	0.201045	+	0.979582i	$0.435566\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.73205	−0.591285
$172$	0	0
$173$	−12.1962	−0.927256	−0.463628	−	0.886030i	$-0.653453\pi$
$173$	−12.1962	−0.927256	−0.463628	+	0.886030i	$0.653453\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−9.46410	−0.711365
$178$	0	0
$179$	−13.3923	−1.00099	−0.500494	−	0.865740i	$-0.666848\pi$
$179$	−13.3923	−1.00099	−0.500494	+	0.865740i	$0.666848\pi$
$180$	0	0
$181$	9.12436	0.678208	0.339104	−	0.940749i	$-0.389876\pi$
$181$	9.12436	0.678208	0.339104	+	0.940749i	$0.389876\pi$
$182$	0	0
$183$	−5.46410	−0.403918
$184$	0	0
$185$	−5.66025	−0.416150
$186$	0	0
$187$	7.46410	0.545829
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	7.85641	0.568470	0.284235	−	0.958755i	$-0.408260\pi$
$191$	7.85641	0.568470	0.284235	+	0.958755i	$0.408260\pi$
$192$	0	0
$193$	−11.4641	−0.825204	−0.412602	−	0.910911i	$-0.635380\pi$
$193$	−11.4641	−0.825204	−0.412602	+	0.910911i	$0.635380\pi$
$194$	0	0
$195$	−4.73205	−0.338869
$196$	0	0
$197$	8.92820	0.636108	0.318054	−	0.948073i	$-0.396971\pi$
$197$	8.92820	0.636108	0.318054	+	0.948073i	$0.396971\pi$
$198$	0	0
$199$	−13.6603	−0.968350	−0.484175	−	0.874971i	$-0.660880\pi$
$199$	−13.6603	−0.968350	−0.484175	+	0.874971i	$0.660880\pi$
$200$	0	0
$201$	28.3923	2.00264
$202$	0	0
$203$	−6.46410	−0.453691
$204$	0	0
$205$	−18.0000	−1.25717
$206$	0	0
$207$	−2.07180	−0.144000
$208$	0	0
$209$	−4.73205	−0.327323
$210$	0	0
$211$	21.7846	1.49971	0.749857	−	0.661600i	$-0.230122\pi$
$211$	21.7846	1.49971	0.749857	+	0.661600i	$0.230122\pi$
$212$	0	0
$213$	−27.8564	−1.90869
$214$	0	0
$215$	21.5885	1.47232
$216$	0	0
$217$	0.267949	0.0181896
$218$	0	0
$219$	26.5885	1.79668
$220$	0	0
$221$	−2.73205	−0.183778
$222$	0	0
$223$	14.6603	0.981723	0.490862	−	0.871238i	$-0.336682\pi$
$223$	14.6603	0.981723	0.490862	+	0.871238i	$0.336682\pi$
$224$	0	0
$225$	−8.92820	−0.595214
$226$	0	0
$227$	−2.53590	−0.168313	−0.0841567	−	0.996453i	$-0.526820\pi$
$227$	−2.53590	−0.168313	−0.0841567	+	0.996453i	$0.526820\pi$
$228$	0	0
$229$	−10.9282	−0.722156	−0.361078	−	0.932536i	$-0.617591\pi$
$229$	−10.9282	−0.722156	−0.361078	+	0.932536i	$0.617591\pi$
$230$	0	0
$231$	7.46410	0.491102
$232$	0	0
$233$	23.7846	1.55818	0.779091	−	0.626911i	$-0.215681\pi$
$233$	23.7846	1.55818	0.779091	+	0.626911i	$0.215681\pi$
$234$	0	0
$235$	9.92820	0.647645
$236$	0	0
$237$	−10.7321	−0.697122
$238$	0	0
$239$	21.4641	1.38840	0.694199	−	0.719783i	$-0.255759\pi$
$239$	21.4641	1.38840	0.694199	+	0.719783i	$0.255759\pi$
$240$	0	0
$241$	11.3397	0.730457	0.365229	−	0.930918i	$-0.380991\pi$
$241$	11.3397	0.730457	0.365229	+	0.930918i	$0.380991\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	1.73205	0.110657
$246$	0	0
$247$	1.73205	0.110208
$248$	0	0
$249$	17.1244	1.08521
$250$	0	0
$251$	−6.58846	−0.415860	−0.207930	−	0.978144i	$-0.566673\pi$
$251$	−6.58846	−0.415860	−0.207930	+	0.978144i	$0.566673\pi$
$252$	0	0
$253$	−1.26795	−0.0797153
$254$	0	0
$255$	12.9282	0.809595
$256$	0	0
$257$	7.26795	0.453362	0.226681	−	0.973969i	$-0.427212\pi$
$257$	7.26795	0.453362	0.226681	+	0.973969i	$0.427212\pi$
$258$	0	0
$259$	−3.26795	−0.203060
$260$	0	0
$261$	−28.8564	−1.78617
$262$	0	0
$263$	−26.3205	−1.62299	−0.811496	−	0.584358i	$-0.801346\pi$
$263$	−26.3205	−1.62299	−0.811496	+	0.584358i	$0.801346\pi$
$264$	0	0
$265$	−17.1962	−1.05635
$266$	0	0
$267$	33.5167	2.05119
$268$	0	0
$269$	−19.3205	−1.17799	−0.588996	−	0.808136i	$-0.700477\pi$
$269$	−19.3205	−1.17799	−0.588996	+	0.808136i	$0.700477\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	−2.73205	−0.165351
$274$	0	0
$275$	−5.46410	−0.329498
$276$	0	0
$277$	−32.4641	−1.95058	−0.975289	−	0.220931i	$-0.929090\pi$
$277$	−32.4641	−1.95058	−0.975289	+	0.220931i	$0.929090\pi$
$278$	0	0
$279$	1.19615	0.0716118
$280$	0	0
$281$	24.5885	1.46682	0.733412	−	0.679784i	$-0.237927\pi$
$281$	24.5885	1.46682	0.733412	+	0.679784i	$0.237927\pi$
$282$	0	0
$283$	14.5359	0.864069	0.432035	−	0.901857i	$-0.357796\pi$
$283$	14.5359	0.864069	0.432035	+	0.901857i	$0.357796\pi$
$284$	0	0
$285$	−8.19615	−0.485498
$286$	0	0
$287$	−10.3923	−0.613438
$288$	0	0
$289$	−9.53590	−0.560935
$290$	0	0
$291$	−44.0526	−2.58241
$292$	0	0
$293$	32.5167	1.89964	0.949822	−	0.312792i	$-0.101264\pi$
$293$	32.5167	1.89964	0.949822	+	0.312792i	$0.101264\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	10.9282	0.634119
$298$	0	0
$299$	0.464102	0.0268397
$300$	0	0
$301$	12.4641	0.718419
$302$	0	0
$303$	27.8564	1.60031
$304$	0	0
$305$	−3.46410	−0.198354
$306$	0	0
$307$	−7.58846	−0.433096	−0.216548	−	0.976272i	$-0.569480\pi$
$307$	−7.58846	−0.433096	−0.216548	+	0.976272i	$0.569480\pi$
$308$	0	0
$309$	−10.9282	−0.621684
$310$	0	0
$311$	6.19615	0.351352	0.175676	−	0.984448i	$-0.443789\pi$
$311$	6.19615	0.351352	0.175676	+	0.984448i	$0.443789\pi$
$312$	0	0
$313$	18.9808	1.07286	0.536428	−	0.843946i	$-0.319773\pi$
$313$	18.9808	1.07286	0.536428	+	0.843946i	$0.319773\pi$
$314$	0	0
$315$	7.73205	0.435652
$316$	0	0
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	−17.6603	−0.988784
$320$	0	0
$321$	27.3205	1.52488
$322$	0	0
$323$	−4.73205	−0.263298
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−10.3923	−0.574696
$328$	0	0
$329$	5.73205	0.316018
$330$	0	0
$331$	−18.7846	−1.03250	−0.516248	−	0.856439i	$-0.672672\pi$
$331$	−18.7846	−1.03250	−0.516248	+	0.856439i	$0.672672\pi$
$332$	0	0
$333$	−14.5885	−0.799443
$334$	0	0
$335$	18.0000	0.983445
$336$	0	0
$337$	−14.4641	−0.787910	−0.393955	−	0.919130i	$-0.628893\pi$
$337$	−14.4641	−0.787910	−0.393955	+	0.919130i	$0.628893\pi$
$338$	0	0
$339$	4.19615	0.227904
$340$	0	0
$341$	0.732051	0.0396428
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.19615	−0.118237
$346$	0	0
$347$	−34.9282	−1.87504	−0.937522	−	0.347926i	$-0.886886\pi$
$347$	−34.9282	−1.87504	−0.937522	+	0.347926i	$0.886886\pi$
$348$	0	0
$349$	−13.5885	−0.727373	−0.363687	−	0.931521i	$-0.618482\pi$
$349$	−13.5885	−0.727373	−0.363687	+	0.931521i	$0.618482\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	25.8564	1.37620	0.688099	−	0.725617i	$-0.258446\pi$
$353$	25.8564	1.37620	0.688099	+	0.725617i	$0.258446\pi$
$354$	0	0
$355$	−17.6603	−0.937309
$356$	0	0
$357$	7.46410	0.395042
$358$	0	0
$359$	−17.5167	−0.924494	−0.462247	−	0.886751i	$-0.652957\pi$
$359$	−17.5167	−0.924494	−0.462247	+	0.886751i	$0.652957\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	0	0
$363$	−9.66025	−0.507032
$364$	0	0
$365$	16.8564	0.882305
$366$	0	0
$367$	29.1244	1.52028	0.760139	−	0.649760i	$-0.225131\pi$
$367$	29.1244	1.52028	0.760139	+	0.649760i	$0.225131\pi$
$368$	0	0
$369$	−46.3923	−2.41509
$370$	0	0
$371$	−9.92820	−0.515447
$372$	0	0
$373$	−21.8564	−1.13168	−0.565841	−	0.824514i	$-0.691448\pi$
$373$	−21.8564	−1.13168	−0.565841	+	0.824514i	$0.691448\pi$
$374$	0	0
$375$	−33.1244	−1.71053
$376$	0	0
$377$	6.46410	0.332918
$378$	0	0
$379$	−4.58846	−0.235693	−0.117847	−	0.993032i	$-0.537599\pi$
$379$	−4.58846	−0.235693	−0.117847	+	0.993032i	$0.537599\pi$
$380$	0	0
$381$	−17.4641	−0.894713
$382$	0	0
$383$	−17.4641	−0.892374	−0.446187	−	0.894940i	$-0.647218\pi$
$383$	−17.4641	−0.892374	−0.446187	+	0.894940i	$0.647218\pi$
$384$	0	0
$385$	4.73205	0.241168
$386$	0	0
$387$	55.6410	2.82839
$388$	0	0
$389$	−21.4641	−1.08827	−0.544137	−	0.838997i	$-0.683143\pi$
$389$	−21.4641	−1.08827	−0.544137	+	0.838997i	$0.683143\pi$
$390$	0	0
$391$	−1.26795	−0.0641229
$392$	0	0
$393$	−17.8564	−0.900737
$394$	0	0
$395$	−6.80385	−0.342339
$396$	0	0
$397$	3.58846	0.180100	0.0900498	−	0.995937i	$-0.471297\pi$
$397$	3.58846	0.180100	0.0900498	+	0.995937i	$0.471297\pi$
$398$	0	0
$399$	−4.73205	−0.236899
$400$	0	0
$401$	8.00000	0.399501	0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000	0.399501	0.199750	+	0.979847i	$0.435987\pi$
$402$	0	0
$403$	−0.267949	−0.0133475
$404$	0	0
$405$	−4.26795	−0.212076
$406$	0	0
$407$	−8.92820	−0.442555
$408$	0	0
$409$	−24.5167	−1.21227	−0.606135	−	0.795361i	$-0.707281\pi$
$409$	−24.5167	−1.21227	−0.606135	+	0.795361i	$0.707281\pi$
$410$	0	0
$411$	19.8564	0.979444
$412$	0	0
$413$	−3.46410	−0.170457
$414$	0	0
$415$	10.8564	0.532920
$416$	0	0
$417$	−50.7846	−2.48693
$418$	0	0
$419$	8.73205	0.426589	0.213294	−	0.976988i	$-0.431581\pi$
$419$	8.73205	0.426589	0.213294	+	0.976988i	$0.431581\pi$
$420$	0	0
$421$	33.9090	1.65262	0.826311	−	0.563214i	$-0.190435\pi$
$421$	33.9090	1.65262	0.826311	+	0.563214i	$0.190435\pi$
$422$	0	0
$423$	25.5885	1.24415
$424$	0	0
$425$	−5.46410	−0.265048
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−7.46410	−0.360370
$430$	0	0
$431$	29.7128	1.43122	0.715608	−	0.698502i	$-0.246150\pi$
$431$	29.7128	1.43122	0.715608	+	0.698502i	$0.246150\pi$
$432$	0	0
$433$	−14.9282	−0.717404	−0.358702	−	0.933452i	$-0.616781\pi$
$433$	−14.9282	−0.717404	−0.358702	+	0.933452i	$0.616781\pi$
$434$	0	0
$435$	−30.5885	−1.46660
$436$	0	0
$437$	0.803848	0.0384532
$438$	0	0
$439$	40.2487	1.92097	0.960483	−	0.278338i	$-0.0897836\pi$
$439$	40.2487	1.92097	0.960483	+	0.278338i	$0.0897836\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	0	0
$443$	−11.7846	−0.559904	−0.279952	−	0.960014i	$-0.590319\pi$
$443$	−11.7846	−0.559904	−0.279952	+	0.960014i	$0.590319\pi$
$444$	0	0
$445$	21.2487	1.00729
$446$	0	0
$447$	48.2487	2.28209
$448$	0	0
$449$	3.60770	0.170258	0.0851288	−	0.996370i	$-0.472870\pi$
$449$	3.60770	0.170258	0.0851288	+	0.996370i	$0.472870\pi$
$450$	0	0
$451$	−28.3923	−1.33694
$452$	0	0
$453$	20.9282	0.983293
$454$	0	0
$455$	−1.73205	−0.0811998
$456$	0	0
$457$	−0.732051	−0.0342439	−0.0171219	−	0.999853i	$-0.505450\pi$
$457$	−0.732051	−0.0342439	−0.0171219	+	0.999853i	$0.505450\pi$
$458$	0	0
$459$	10.9282	0.510085
$460$	0	0
$461$	−2.39230	−0.111421	−0.0557104	−	0.998447i	$-0.517742\pi$
$461$	−2.39230	−0.111421	−0.0557104	+	0.998447i	$0.517742\pi$
$462$	0	0
$463$	−8.58846	−0.399139	−0.199570	−	0.979884i	$-0.563954\pi$
$463$	−8.58846	−0.399139	−0.199570	+	0.979884i	$0.563954\pi$
$464$	0	0
$465$	1.26795	0.0587997
$466$	0	0
$467$	6.58846	0.304877	0.152439	−	0.988313i	$-0.451287\pi$
$467$	6.58846	0.304877	0.152439	+	0.988313i	$0.451287\pi$
$468$	0	0
$469$	10.3923	0.479872
$470$	0	0
$471$	−37.3205	−1.71964
$472$	0	0
$473$	34.0526	1.56574
$474$	0	0
$475$	3.46410	0.158944
$476$	0	0
$477$	−44.3205	−2.02930
$478$	0	0
$479$	15.8756	0.725377	0.362688	−	0.931910i	$-0.381859\pi$
$479$	15.8756	0.725377	0.362688	+	0.931910i	$0.381859\pi$
$480$	0	0
$481$	3.26795	0.149006
$482$	0	0
$483$	−1.26795	−0.0576937
$484$	0	0
$485$	−27.9282	−1.26815
$486$	0	0
$487$	−14.3923	−0.652178	−0.326089	−	0.945339i	$-0.605731\pi$
$487$	−14.3923	−0.652178	−0.326089	+	0.945339i	$0.605731\pi$
$488$	0	0
$489$	25.8564	1.16927
$490$	0	0
$491$	−9.32051	−0.420629	−0.210314	−	0.977634i	$-0.567449\pi$
$491$	−9.32051	−0.420629	−0.210314	+	0.977634i	$0.567449\pi$
$492$	0	0
$493$	−17.6603	−0.795378
$494$	0	0
$495$	21.1244	0.949469
$496$	0	0
$497$	−10.1962	−0.457360
$498$	0	0
$499$	−29.5167	−1.32135	−0.660674	−	0.750673i	$-0.729729\pi$
$499$	−29.5167	−1.32135	−0.660674	+	0.750673i	$0.729729\pi$
$500$	0	0
$501$	14.1962	0.634237
$502$	0	0
$503$	32.6410	1.45539	0.727695	−	0.685900i	$-0.240591\pi$
$503$	32.6410	1.45539	0.727695	+	0.685900i	$0.240591\pi$
$504$	0	0
$505$	17.6603	0.785871
$506$	0	0
$507$	2.73205	0.121335
$508$	0	0
$509$	−26.5167	−1.17533	−0.587665	−	0.809104i	$-0.699953\pi$
$509$	−26.5167	−1.17533	−0.587665	+	0.809104i	$0.699953\pi$
$510$	0	0
$511$	9.73205	0.430521
$512$	0	0
$513$	−6.92820	−0.305888
$514$	0	0
$515$	−6.92820	−0.305293
$516$	0	0
$517$	15.6603	0.688737
$518$	0	0
$519$	−33.3205	−1.46261
$520$	0	0
$521$	24.7846	1.08583	0.542917	−	0.839787i	$-0.317320\pi$
$521$	24.7846	1.08583	0.542917	+	0.839787i	$0.317320\pi$
$522$	0	0
$523$	−35.8564	−1.56789	−0.783946	−	0.620830i	$-0.786796\pi$
$523$	−35.8564	−1.56789	−0.783946	+	0.620830i	$0.786796\pi$
$524$	0	0
$525$	−5.46410	−0.238473
$526$	0	0
$527$	0.732051	0.0318886
$528$	0	0
$529$	−22.7846	−0.990635
$530$	0	0
$531$	−15.4641	−0.671085
$532$	0	0
$533$	10.3923	0.450141
$534$	0	0
$535$	17.3205	0.748831
$536$	0	0
$537$	−36.5885	−1.57891
$538$	0	0
$539$	2.73205	0.117678
$540$	0	0
$541$	−17.5167	−0.753100	−0.376550	−	0.926396i	$-0.622890\pi$
$541$	−17.5167	−0.753100	−0.376550	+	0.926396i	$0.622890\pi$
$542$	0	0
$543$	24.9282	1.06977
$544$	0	0
$545$	−6.58846	−0.282219
$546$	0	0
$547$	−36.3205	−1.55295	−0.776476	−	0.630146i	$-0.782995\pi$
$547$	−36.3205	−1.55295	−0.776476	+	0.630146i	$0.782995\pi$
$548$	0	0
$549$	−8.92820	−0.381046
$550$	0	0
$551$	11.1962	0.476972
$552$	0	0
$553$	−3.92820	−0.167044
$554$	0	0
$555$	−15.4641	−0.656415
$556$	0	0
$557$	27.7128	1.17423	0.587115	−	0.809504i	$-0.300264\pi$
$557$	27.7128	1.17423	0.587115	+	0.809504i	$0.300264\pi$
$558$	0	0
$559$	−12.4641	−0.527175
$560$	0	0
$561$	20.3923	0.860964
$562$	0	0
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	0	0
$565$	2.66025	0.111918
$566$	0	0
$567$	−2.46410	−0.103483
$568$	0	0
$569$	17.5359	0.735143	0.367572	−	0.929995i	$-0.380189\pi$
$569$	17.5359	0.735143	0.367572	+	0.929995i	$0.380189\pi$
$570$	0	0
$571$	46.1769	1.93244	0.966222	−	0.257712i	$-0.0829685\pi$
$571$	46.1769	1.93244	0.966222	+	0.257712i	$0.0829685\pi$
$572$	0	0
$573$	21.4641	0.896676
$574$	0	0
$575$	0.928203	0.0387088
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	−31.3205	−1.30164
$580$	0	0
$581$	6.26795	0.260038
$582$	0	0
$583$	−27.1244	−1.12338
$584$	0	0
$585$	−7.73205	−0.319681
$586$	0	0
$587$	38.6603	1.59568	0.797840	−	0.602870i	$-0.205976\pi$
$587$	38.6603	1.59568	0.797840	+	0.602870i	$0.205976\pi$
$588$	0	0
$589$	−0.464102	−0.0191230
$590$	0	0
$591$	24.3923	1.00337
$592$	0	0
$593$	37.0526	1.52157	0.760783	−	0.649006i	$-0.224815\pi$
$593$	37.0526	1.52157	0.760783	+	0.649006i	$0.224815\pi$
$594$	0	0
$595$	4.73205	0.193995
$596$	0	0
$597$	−37.3205	−1.52743
$598$	0	0
$599$	15.7846	0.644942	0.322471	−	0.946579i	$-0.395487\pi$
$599$	15.7846	0.644942	0.322471	+	0.946579i	$0.395487\pi$
$600$	0	0
$601$	−3.85641	−0.157306	−0.0786531	−	0.996902i	$-0.525062\pi$
$601$	−3.85641	−0.157306	−0.0786531	+	0.996902i	$0.525062\pi$
$602$	0	0
$603$	46.3923	1.88924
$604$	0	0
$605$	−6.12436	−0.248990
$606$	0	0
$607$	38.1962	1.55033	0.775167	−	0.631756i	$-0.217666\pi$
$607$	38.1962	1.55033	0.775167	+	0.631756i	$0.217666\pi$
$608$	0	0
$609$	−17.6603	−0.715630
$610$	0	0
$611$	−5.73205	−0.231894
$612$	0	0
$613$	33.8564	1.36745	0.683724	−	0.729741i	$-0.260359\pi$
$613$	33.8564	1.36745	0.683724	+	0.729741i	$0.260359\pi$
$614$	0	0
$615$	−49.1769	−1.98300
$616$	0	0
$617$	−40.6410	−1.63615	−0.818073	−	0.575115i	$-0.804957\pi$
$617$	−40.6410	−1.63615	−0.818073	+	0.575115i	$0.804957\pi$
$618$	0	0
$619$	−35.7128	−1.43542	−0.717710	−	0.696343i	$-0.754809\pi$
$619$	−35.7128	−1.43542	−0.717710	+	0.696343i	$0.754809\pi$
$620$	0	0
$621$	−1.85641	−0.0744950
$622$	0	0
$623$	12.2679	0.491505
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−12.9282	−0.516303
$628$	0	0
$629$	−8.92820	−0.355991
$630$	0	0
$631$	29.7128	1.18285	0.591424	−	0.806361i	$-0.298566\pi$
$631$	29.7128	1.18285	0.591424	+	0.806361i	$0.298566\pi$
$632$	0	0
$633$	59.5167	2.36557
$634$	0	0
$635$	−11.0718	−0.439371
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−45.5167	−1.80061
$640$	0	0
$641$	−25.6410	−1.01276	−0.506380	−	0.862311i	$-0.669017\pi$
$641$	−25.6410	−1.01276	−0.506380	+	0.862311i	$0.669017\pi$
$642$	0	0
$643$	45.1769	1.78160	0.890802	−	0.454392i	$-0.150143\pi$
$643$	45.1769	1.78160	0.890802	+	0.454392i	$0.150143\pi$
$644$	0	0
$645$	58.9808	2.32237
$646$	0	0
$647$	−6.58846	−0.259019	−0.129509	−	0.991578i	$-0.541340\pi$
$647$	−6.58846	−0.259019	−0.129509	+	0.991578i	$0.541340\pi$
$648$	0	0
$649$	−9.46410	−0.371498
$650$	0	0
$651$	0.732051	0.0286913
$652$	0	0
$653$	19.8564	0.777041	0.388521	−	0.921440i	$-0.372986\pi$
$653$	19.8564	0.777041	0.388521	+	0.921440i	$0.372986\pi$
$654$	0	0
$655$	−11.3205	−0.442329
$656$	0	0
$657$	43.4449	1.69495
$658$	0	0
$659$	−11.9282	−0.464657	−0.232328	−	0.972637i	$-0.574634\pi$
$659$	−11.9282	−0.464657	−0.232328	+	0.972637i	$0.574634\pi$
$660$	0	0
$661$	20.8038	0.809176	0.404588	−	0.914499i	$-0.367415\pi$
$661$	20.8038	0.809176	0.404588	+	0.914499i	$0.367415\pi$
$662$	0	0
$663$	−7.46410	−0.289882
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	3.00000	0.116160
$668$	0	0
$669$	40.0526	1.54852
$670$	0	0
$671$	−5.46410	−0.210939
$672$	0	0
$673$	6.60770	0.254708	0.127354	−	0.991857i	$-0.459352\pi$
$673$	6.60770	0.254708	0.127354	+	0.991857i	$0.459352\pi$
$674$	0	0
$675$	−8.00000	−0.307920
$676$	0	0
$677$	30.9808	1.19069	0.595344	−	0.803471i	$-0.297016\pi$
$677$	30.9808	1.19069	0.595344	+	0.803471i	$0.297016\pi$
$678$	0	0
$679$	−16.1244	−0.618796
$680$	0	0
$681$	−6.92820	−0.265489
$682$	0	0
$683$	−5.46410	−0.209078	−0.104539	−	0.994521i	$-0.533337\pi$
$683$	−5.46410	−0.209078	−0.104539	+	0.994521i	$0.533337\pi$
$684$	0	0
$685$	12.5885	0.480980
$686$	0	0
$687$	−29.8564	−1.13909
$688$	0	0
$689$	9.92820	0.378234
$690$	0	0
$691$	−49.3013	−1.87551	−0.937754	−	0.347299i	$-0.887099\pi$
$691$	−49.3013	−1.87551	−0.937754	+	0.347299i	$0.887099\pi$
$692$	0	0
$693$	12.1962	0.463294
$694$	0	0
$695$	−32.1962	−1.22127
$696$	0	0
$697$	−28.3923	−1.07544
$698$	0	0
$699$	64.9808	2.45780
$700$	0	0
$701$	−6.32051	−0.238722	−0.119361	−	0.992851i	$-0.538085\pi$
$701$	−6.32051	−0.238722	−0.119361	+	0.992851i	$0.538085\pi$
$702$	0	0
$703$	5.66025	0.213481
$704$	0	0
$705$	27.1244	1.02156
$706$	0	0
$707$	10.1962	0.383466
$708$	0	0
$709$	46.0526	1.72954	0.864770	−	0.502168i	$-0.167464\pi$
$709$	46.0526	1.72954	0.864770	+	0.502168i	$0.167464\pi$
$710$	0	0
$711$	−17.5359	−0.657648
$712$	0	0
$713$	−0.124356	−0.00465716
$714$	0	0
$715$	−4.73205	−0.176969
$716$	0	0
$717$	58.6410	2.18999
$718$	0	0
$719$	19.2679	0.718573	0.359287	−	0.933227i	$-0.383020\pi$
$719$	19.2679	0.718573	0.359287	+	0.933227i	$0.383020\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	30.9808	1.15219
$724$	0	0
$725$	12.9282	0.480141
$726$	0	0
$727$	6.78461	0.251627	0.125814	−	0.992054i	$-0.459846\pi$
$727$	6.78461	0.251627	0.125814	+	0.992054i	$0.459846\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	34.0526	1.25948
$732$	0	0
$733$	45.3013	1.67324	0.836620	−	0.547783i	$-0.184528\pi$
$733$	45.3013	1.67324	0.836620	+	0.547783i	$0.184528\pi$
$734$	0	0
$735$	4.73205	0.174544
$736$	0	0
$737$	28.3923	1.04584
$738$	0	0
$739$	−5.26795	−0.193785	−0.0968923	−	0.995295i	$-0.530890\pi$
$739$	−5.26795	−0.193785	−0.0968923	+	0.995295i	$0.530890\pi$
$740$	0	0
$741$	4.73205	0.173836
$742$	0	0
$743$	37.8564	1.38882	0.694408	−	0.719581i	$-0.255666\pi$
$743$	37.8564	1.38882	0.694408	+	0.719581i	$0.255666\pi$
$744$	0	0
$745$	30.5885	1.12067
$746$	0	0
$747$	27.9808	1.02376
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	−3.92820	−0.143342	−0.0716711	−	0.997428i	$-0.522833\pi$
$751$	−3.92820	−0.143342	−0.0716711	+	0.997428i	$0.522833\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	13.2679	0.482870
$756$	0	0
$757$	27.5359	1.00081	0.500405	−	0.865792i	$-0.333185\pi$
$757$	27.5359	1.00081	0.500405	+	0.865792i	$0.333185\pi$
$758$	0	0
$759$	−3.46410	−0.125739
$760$	0	0
$761$	36.2679	1.31471	0.657356	−	0.753580i	$-0.271675\pi$
$761$	36.2679	1.31471	0.657356	+	0.753580i	$0.271675\pi$
$762$	0	0
$763$	−3.80385	−0.137709
$764$	0	0
$765$	21.1244	0.763753
$766$	0	0
$767$	3.46410	0.125081
$768$	0	0
$769$	11.8756	0.428247	0.214123	−	0.976807i	$-0.431311\pi$
$769$	11.8756	0.428247	0.214123	+	0.976807i	$0.431311\pi$
$770$	0	0
$771$	19.8564	0.715111
$772$	0	0
$773$	12.5359	0.450885	0.225442	−	0.974256i	$-0.427617\pi$
$773$	12.5359	0.450885	0.225442	+	0.974256i	$0.427617\pi$
$774$	0	0
$775$	−0.535898	−0.0192500
$776$	0	0
$777$	−8.92820	−0.320298
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	−27.8564	−0.996781
$782$	0	0
$783$	−25.8564	−0.924033
$784$	0	0
$785$	−23.6603	−0.844471
$786$	0	0
$787$	−32.5167	−1.15909	−0.579547	−	0.814939i	$-0.696770\pi$
$787$	−32.5167	−1.15909	−0.579547	+	0.814939i	$0.696770\pi$
$788$	0	0
$789$	−71.9090	−2.56003
$790$	0	0
$791$	1.53590	0.0546103
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−46.9808	−1.66624
$796$	0	0
$797$	26.7846	0.948760	0.474380	−	0.880320i	$-0.342672\pi$
$797$	26.7846	0.948760	0.474380	+	0.880320i	$0.342672\pi$
$798$	0	0
$799$	15.6603	0.554020
$800$	0	0
$801$	54.7654	1.93504
$802$	0	0
$803$	26.5885	0.938286
$804$	0	0
$805$	−0.803848	−0.0283319
$806$	0	0
$807$	−52.7846	−1.85811
$808$	0	0
$809$	40.0333	1.40750	0.703748	−	0.710449i	$-0.251508\pi$
$809$	40.0333	1.40750	0.703748	+	0.710449i	$0.251508\pi$
$810$	0	0
$811$	12.7846	0.448928	0.224464	−	0.974482i	$-0.427937\pi$
$811$	12.7846	0.448928	0.224464	+	0.974482i	$0.427937\pi$
$812$	0	0
$813$	−32.7846	−1.14981
$814$	0	0
$815$	16.3923	0.574197
$816$	0	0
$817$	−21.5885	−0.755285
$818$	0	0
$819$	−4.46410	−0.155988
$820$	0	0
$821$	14.6795	0.512318	0.256159	−	0.966635i	$-0.417543\pi$
$821$	14.6795	0.512318	0.256159	+	0.966635i	$0.417543\pi$
$822$	0	0
$823$	47.8564	1.66817	0.834085	−	0.551636i	$-0.185996\pi$
$823$	47.8564	1.66817	0.834085	+	0.551636i	$0.185996\pi$
$824$	0	0
$825$	−14.9282	−0.519733
$826$	0	0
$827$	37.1769	1.29277	0.646384	−	0.763012i	$-0.276280\pi$
$827$	37.1769	1.29277	0.646384	+	0.763012i	$0.276280\pi$
$828$	0	0
$829$	−44.0526	−1.53001	−0.765004	−	0.644025i	$-0.777263\pi$
$829$	−44.0526	−1.53001	−0.765004	+	0.644025i	$0.777263\pi$
$830$	0	0
$831$	−88.6936	−3.07675
$832$	0	0
$833$	2.73205	0.0946600
$834$	0	0
$835$	9.00000	0.311458
$836$	0	0
$837$	1.07180	0.0370467
$838$	0	0
$839$	41.4641	1.43150	0.715750	−	0.698357i	$-0.246085\pi$
$839$	41.4641	1.43150	0.715750	+	0.698357i	$0.246085\pi$
$840$	0	0
$841$	12.7846	0.440849
$842$	0	0
$843$	67.1769	2.31370
$844$	0	0
$845$	1.73205	0.0595844
$846$	0	0
$847$	−3.53590	−0.121495
$848$	0	0
$849$	39.7128	1.36294
$850$	0	0
$851$	1.51666	0.0519905
$852$	0	0
$853$	50.1244	1.71623	0.858113	−	0.513462i	$-0.171637\pi$
$853$	50.1244	1.71623	0.858113	+	0.513462i	$0.171637\pi$
$854$	0	0
$855$	−13.3923	−0.458007
$856$	0	0
$857$	−35.4641	−1.21143	−0.605715	−	0.795681i	$-0.707113\pi$
$857$	−35.4641	−1.21143	−0.605715	+	0.795681i	$0.707113\pi$
$858$	0	0
$859$	47.5692	1.62304	0.811520	−	0.584324i	$-0.198640\pi$
$859$	47.5692	1.62304	0.811520	+	0.584324i	$0.198640\pi$
$860$	0	0
$861$	−28.3923	−0.967607
$862$	0	0
$863$	−33.0718	−1.12578	−0.562889	−	0.826533i	$-0.690310\pi$
$863$	−33.0718	−1.12578	−0.562889	+	0.826533i	$0.690310\pi$
$864$	0	0
$865$	−21.1244	−0.718250
$866$	0	0
$867$	−26.0526	−0.884791
$868$	0	0
$869$	−10.7321	−0.364060
$870$	0	0
$871$	−10.3923	−0.352130
$872$	0	0
$873$	−71.9808	−2.43618
$874$	0	0
$875$	−12.1244	−0.409878
$876$	0	0
$877$	−9.51666	−0.321355	−0.160677	−	0.987007i	$-0.551368\pi$
$877$	−9.51666	−0.321355	−0.160677	+	0.987007i	$0.551368\pi$
$878$	0	0
$879$	88.8372	2.99640
$880$	0	0
$881$	49.3205	1.66165	0.830825	−	0.556534i	$-0.187869\pi$
$881$	49.3205	1.66165	0.830825	+	0.556534i	$0.187869\pi$
$882$	0	0
$883$	−44.2487	−1.48909	−0.744544	−	0.667574i	$-0.767333\pi$
$883$	−44.2487	−1.48909	−0.744544	+	0.667574i	$0.767333\pi$
$884$	0	0
$885$	−16.3923	−0.551021
$886$	0	0
$887$	−51.0333	−1.71353	−0.856766	−	0.515706i	$-0.827530\pi$
$887$	−51.0333	−1.71353	−0.856766	+	0.515706i	$0.827530\pi$
$888$	0	0
$889$	−6.39230	−0.214391
$890$	0	0
$891$	−6.73205	−0.225532
$892$	0	0
$893$	−9.92820	−0.332235
$894$	0	0
$895$	−23.1962	−0.775362
$896$	0	0
$897$	1.26795	0.0423356
$898$	0	0
$899$	−1.73205	−0.0577671
$900$	0	0
$901$	−27.1244	−0.903643
$902$	0	0
$903$	34.0526	1.13320
$904$	0	0
$905$	15.8038	0.525338
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	0	0
$909$	45.5167	1.50969
$910$	0	0
$911$	43.2487	1.43289	0.716447	−	0.697642i	$-0.245767\pi$
$911$	43.2487	1.43289	0.716447	+	0.697642i	$0.245767\pi$
$912$	0	0
$913$	17.1244	0.566733
$914$	0	0
$915$	−9.46410	−0.312874
$916$	0	0
$917$	−6.53590	−0.215834
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−20.7321	−0.683144
$922$	0	0
$923$	10.1962	0.335610
$924$	0	0
$925$	6.53590	0.214899
$926$	0	0
$927$	−17.8564	−0.586481
$928$	0	0
$929$	33.4449	1.09729	0.548645	−	0.836055i	$-0.315144\pi$
$929$	33.4449	1.09729	0.548645	+	0.836055i	$0.315144\pi$
$930$	0	0
$931$	−1.73205	−0.0567657
$932$	0	0
$933$	16.9282	0.554204
$934$	0	0
$935$	12.9282	0.422797
$936$	0	0
$937$	26.8372	0.876732	0.438366	−	0.898797i	$-0.355557\pi$
$937$	26.8372	0.876732	0.438366	+	0.898797i	$0.355557\pi$
$938$	0	0
$939$	51.8564	1.69227
$940$	0	0
$941$	−18.8038	−0.612988	−0.306494	−	0.951873i	$-0.599156\pi$
$941$	−18.8038	−0.612988	−0.306494	+	0.951873i	$0.599156\pi$
$942$	0	0
$943$	4.82309	0.157061
$944$	0	0
$945$	6.92820	0.225374
$946$	0	0
$947$	6.67949	0.217054	0.108527	−	0.994093i	$-0.465387\pi$
$947$	6.67949	0.217054	0.108527	+	0.994093i	$0.465387\pi$
$948$	0	0
$949$	−9.73205	−0.315916
$950$	0	0
$951$	−76.4974	−2.48060
$952$	0	0
$953$	−24.4641	−0.792470	−0.396235	−	0.918149i	$-0.629683\pi$
$953$	−24.4641	−0.792470	−0.396235	+	0.918149i	$0.629683\pi$
$954$	0	0
$955$	13.6077	0.440335
$956$	0	0
$957$	−48.2487	−1.55966
$958$	0	0
$959$	7.26795	0.234694
$960$	0	0
$961$	−30.9282	−0.997684
$962$	0	0
$963$	44.6410	1.43854
$964$	0	0
$965$	−19.8564	−0.639200
$966$	0	0
$967$	8.98076	0.288802	0.144401	−	0.989519i	$-0.453875\pi$
$967$	8.98076	0.288802	0.144401	+	0.989519i	$0.453875\pi$
$968$	0	0
$969$	−12.9282	−0.415314
$970$	0	0
$971$	25.6077	0.821790	0.410895	−	0.911683i	$-0.365216\pi$
$971$	25.6077	0.821790	0.410895	+	0.911683i	$0.365216\pi$
$972$	0	0
$973$	−18.5885	−0.595919
$974$	0	0
$975$	5.46410	0.174991
$976$	0	0
$977$	−20.5885	−0.658683	−0.329342	−	0.944211i	$-0.606827\pi$
$977$	−20.5885	−0.658683	−0.329342	+	0.944211i	$0.606827\pi$
$978$	0	0
$979$	33.5167	1.07120
$980$	0	0
$981$	−16.9808	−0.542154
$982$	0	0
$983$	27.7321	0.884515	0.442258	−	0.896888i	$-0.354178\pi$
$983$	27.7321	0.884515	0.442258	+	0.896888i	$0.354178\pi$
$984$	0	0
$985$	15.4641	0.492727
$986$	0	0
$987$	15.6603	0.498471
$988$	0	0
$989$	−5.78461	−0.183940
$990$	0	0
$991$	18.3923	0.584251	0.292125	−	0.956380i	$-0.405638\pi$
$991$	18.3923	0.584251	0.292125	+	0.956380i	$0.405638\pi$
$992$	0	0
$993$	−51.3205	−1.62861
$994$	0	0
$995$	−23.6603	−0.750080
$996$	0	0
$997$	−8.39230	−0.265787	−0.132893	−	0.991130i	$-0.542427\pi$
$997$	−8.39230	−0.265787	−0.132893	+	0.991130i	$0.542427\pi$
$998$	0	0
$999$	−13.0718	−0.413573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.r.1.2		2
4.3	odd	2		728.2.a.f.1.1	✓	2
8.3	odd	2		5824.2.a.bp.1.2		2
8.5	even	2		5824.2.a.bi.1.1		2
12.11	even	2		6552.2.a.bf.1.1		2
28.27	even	2		5096.2.a.o.1.2		2
52.51	odd	2		9464.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.f.1.1	✓	2	4.3	odd	2
1456.2.a.r.1.2		2	1.1	even	1	trivial
5096.2.a.o.1.2		2	28.27	even	2
5824.2.a.bi.1.1		2	8.5	even	2
5824.2.a.bp.1.2		2	8.3	odd	2
6552.2.a.bf.1.1		2	12.11	even	2
9464.2.a.n.1.1		2	52.51	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} +1.73205 q^{5} +1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q+2.73205 q^{3} +1.73205 q^{5} +1.00000 q^{7} +4.46410 q^{9} +2.73205 q^{11} -1.00000 q^{13} +4.73205 q^{15} +2.73205 q^{17} -1.73205 q^{19} +2.73205 q^{21} -0.464102 q^{23} -2.00000 q^{25} +4.00000 q^{27} -6.46410 q^{29} +0.267949 q^{31} +7.46410 q^{33} +1.73205 q^{35} -3.26795 q^{37} -2.73205 q^{39} -10.3923 q^{41} +12.4641 q^{43} +7.73205 q^{45} +5.73205 q^{47} +1.00000 q^{49} +7.46410 q^{51} -9.92820 q^{53} +4.73205 q^{55} -4.73205 q^{57} -3.46410 q^{59} -2.00000 q^{61} +4.46410 q^{63} -1.73205 q^{65} +10.3923 q^{67} -1.26795 q^{69} -10.1962 q^{71} +9.73205 q^{73} -5.46410 q^{75} +2.73205 q^{77} -3.92820 q^{79} -2.46410 q^{81} +6.26795 q^{83} +4.73205 q^{85} -17.6603 q^{87} +12.2679 q^{89} -1.00000 q^{91} +0.732051 q^{93} -3.00000 q^{95} -16.1244 q^{97} +12.1962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 6 q^{15} + 2 q^{17} + 2 q^{21} + 6 q^{23} - 4 q^{25} + 8 q^{27} - 6 q^{29} + 4 q^{31} + 8 q^{33} - 10 q^{37} - 2 q^{39} + 18 q^{43} + 12 q^{45} + 8 q^{47} + 2 q^{49} + 8 q^{51} - 6 q^{53} + 6 q^{55} - 6 q^{57} - 4 q^{61} + 2 q^{63} - 6 q^{69} - 10 q^{71} + 16 q^{73} - 4 q^{75} + 2 q^{77} + 6 q^{79} + 2 q^{81} + 16 q^{83} + 6 q^{85} - 18 q^{87} + 28 q^{89} - 2 q^{91} - 2 q^{93} - 6 q^{95} - 8 q^{97} + 14 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 6 * q^15 + 2 * q^17 + 2 * q^21 + 6 * q^23 - 4 * q^25 + 8 * q^27 - 6 * q^29 + 4 * q^31 + 8 * q^33 - 10 * q^37 - 2 * q^39 + 18 * q^43 + 12 * q^45 + 8 * q^47 + 2 * q^49 + 8 * q^51 - 6 * q^53 + 6 * q^55 - 6 * q^57 - 4 * q^61 + 2 * q^63 - 6 * q^69 - 10 * q^71 + 16 * q^73 - 4 * q^75 + 2 * q^77 + 6 * q^79 + 2 * q^81 + 16 * q^83 + 6 * q^85 - 18 * q^87 + 28 * q^89 - 2 * q^91 - 2 * q^93 - 6 * q^95 - 8 * q^97 + 14 * q^99

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