LMFDB - Embedded newform 1456.2.a.n.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:	$1$
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 364)
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)

$f(q)$	$=$	$q+0.732051 q^{3} +1.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} +1.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} -4.73205 q^{11} +1.00000 q^{13} +1.26795 q^{15} -2.19615 q^{17} -7.19615 q^{19} -0.732051 q^{21} -3.00000 q^{23} -2.00000 q^{25} -4.00000 q^{27} +0.464102 q^{29} -1.19615 q^{31} -3.46410 q^{33} -1.73205 q^{35} +4.19615 q^{37} +0.732051 q^{39} +3.46410 q^{41} +7.00000 q^{43} -4.26795 q^{45} -7.73205 q^{47} +1.00000 q^{49} -1.60770 q^{51} -9.92820 q^{53} -8.19615 q^{55} -5.26795 q^{57} +10.3923 q^{59} -10.0000 q^{61} +2.46410 q^{63} +1.73205 q^{65} +14.3923 q^{67} -2.19615 q^{69} +1.26795 q^{71} -9.19615 q^{73} -1.46410 q^{75} +4.73205 q^{77} +11.3923 q^{79} +4.46410 q^{81} +0.803848 q^{83} -3.80385 q^{85} +0.339746 q^{87} +11.1962 q^{89} -1.00000 q^{91} -0.875644 q^{93} -12.4641 q^{95} +2.80385 q^{97} +11.6603 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 6 q^{23} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 8 q^{31} - 2 q^{37} - 2 q^{39} + 14 q^{43} - 12 q^{45} - 12 q^{47} + 2 q^{49} - 24 q^{51} - 6 q^{53} - 6 q^{55} - 14 q^{57} - 20 q^{61} - 2 q^{63} + 8 q^{67} + 6 q^{69} + 6 q^{71} - 8 q^{73} + 4 q^{75} + 6 q^{77} + 2 q^{79} + 2 q^{81} + 12 q^{83} - 18 q^{85} + 18 q^{87} + 12 q^{89} - 2 q^{91} - 26 q^{93} - 18 q^{95} + 16 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 - 6 * q^23 - 4 * q^25 - 8 * q^27 - 6 * q^29 + 8 * q^31 - 2 * q^37 - 2 * q^39 + 14 * q^43 - 12 * q^45 - 12 * q^47 + 2 * q^49 - 24 * q^51 - 6 * q^53 - 6 * q^55 - 14 * q^57 - 20 * q^61 - 2 * q^63 + 8 * q^67 + 6 * q^69 + 6 * q^71 - 8 * q^73 + 4 * q^75 + 6 * q^77 + 2 * q^79 + 2 * q^81 + 12 * q^83 - 18 * q^85 + 18 * q^87 + 12 * q^89 - 2 * q^91 - 26 * q^93 - 18 * q^95 + 16 * q^97 + 6 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	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
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.26795	0.327383
$16$	0	0
$17$	−2.19615	−0.532645	−0.266323	−	0.963884i	$-0.585809\pi$
$17$	−2.19615	−0.532645	−0.266323	+	0.963884i	$0.585809\pi$
$18$	0	0
$19$	−7.19615	−1.65091	−0.825455	−	0.564467i	$-0.809082\pi$
$19$	−7.19615	−1.65091	−0.825455	+	0.564467i	$0.809082\pi$
$20$	0	0
$21$	−0.732051	−0.159747
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	0.464102	0.0861815	0.0430908	−	0.999071i	$-0.486280\pi$
$29$	0.464102	0.0861815	0.0430908	+	0.999071i	$0.486280\pi$
$30$	0	0
$31$	−1.19615	−0.214835	−0.107418	−	0.994214i	$-0.534258\pi$
$31$	−1.19615	−0.214835	−0.107418	+	0.994214i	$0.534258\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	−1.73205	−0.292770
$36$	0	0
$37$	4.19615	0.689843	0.344922	−	0.938631i	$-0.387905\pi$
$37$	4.19615	0.689843	0.344922	+	0.938631i	$0.387905\pi$
$38$	0	0
$39$	0.732051	0.117222
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	−4.26795	−0.636228
$46$	0	0
$47$	−7.73205	−1.12784	−0.563918	−	0.825831i	$-0.690707\pi$
$47$	−7.73205	−1.12784	−0.563918	+	0.825831i	$0.690707\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.60770	−0.225122
$52$	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
$55$	−8.19615	−1.10517
$56$	0	0
$57$	−5.26795	−0.697757
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	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
$63$	2.46410	0.310448
$64$	0	0
$65$	1.73205	0.214834
$66$	0	0
$67$	14.3923	1.75830	0.879150	−	0.476545i	$-0.158111\pi$
$67$	14.3923	1.75830	0.879150	+	0.476545i	$0.158111\pi$
$68$	0	0
$69$	−2.19615	−0.264386
$70$	0	0
$71$	1.26795	0.150478	0.0752389	−	0.997166i	$-0.476028\pi$
$71$	1.26795	0.150478	0.0752389	+	0.997166i	$0.476028\pi$
$72$	0	0
$73$	−9.19615	−1.07633	−0.538164	−	0.842840i	$-0.680882\pi$
$73$	−9.19615	−1.07633	−0.538164	+	0.842840i	$0.680882\pi$
$74$	0	0
$75$	−1.46410	−0.169060
$76$	0	0
$77$	4.73205	0.539267
$78$	0	0
$79$	11.3923	1.28173	0.640867	−	0.767652i	$-0.278575\pi$
$79$	11.3923	1.28173	0.640867	+	0.767652i	$0.278575\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	0.803848	0.0882337	0.0441169	−	0.999026i	$-0.485953\pi$
$83$	0.803848	0.0882337	0.0441169	+	0.999026i	$0.485953\pi$
$84$	0	0
$85$	−3.80385	−0.412585
$86$	0	0
$87$	0.339746	0.0364246
$88$	0	0
$89$	11.1962	1.18679	0.593395	−	0.804911i	$-0.297787\pi$
$89$	11.1962	1.18679	0.593395	+	0.804911i	$0.297787\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.875644	−0.0908001
$94$	0	0
$95$	−12.4641	−1.27879
$96$	0	0
$97$	2.80385	0.284688	0.142344	−	0.989817i	$-0.454536\pi$
$97$	2.80385	0.284688	0.142344	+	0.989817i	$0.454536\pi$
$98$	0	0
$99$	11.6603	1.17190
$100$	0	0
$101$	15.1244	1.50493	0.752465	−	0.658632i	$-0.228865\pi$
$101$	15.1244	1.50493	0.752465	+	0.658632i	$0.228865\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−1.26795	−0.123739
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−18.1962	−1.74288	−0.871438	−	0.490506i	$-0.836812\pi$
$109$	−18.1962	−1.74288	−0.871438	+	0.490506i	$0.836812\pi$
$110$	0	0
$111$	3.07180	0.291562
$112$	0	0
$113$	−18.4641	−1.73696	−0.868478	−	0.495727i	$-0.834902\pi$
$113$	−18.4641	−1.73696	−0.868478	+	0.495727i	$0.834902\pi$
$114$	0	0
$115$	−5.19615	−0.484544
$116$	0	0
$117$	−2.46410	−0.227806
$118$	0	0
$119$	2.19615	0.201321
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	2.53590	0.228654
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	14.3923	1.27711	0.638555	−	0.769576i	$-0.279532\pi$
$127$	14.3923	1.27711	0.638555	+	0.769576i	$0.279532\pi$
$128$	0	0
$129$	5.12436	0.451174
$130$	0	0
$131$	−14.5359	−1.27001	−0.635004	−	0.772509i	$-0.719001\pi$
$131$	−14.5359	−1.27001	−0.635004	+	0.772509i	$0.719001\pi$
$132$	0	0
$133$	7.19615	0.623986
$134$	0	0
$135$	−6.92820	−0.596285
$136$	0	0
$137$	12.5885	1.07550	0.537752	−	0.843103i	$-0.319274\pi$
$137$	12.5885	1.07550	0.537752	+	0.843103i	$0.319274\pi$
$138$	0	0
$139$	0.196152	0.0166374	0.00831872	−	0.999965i	$-0.497352\pi$
$139$	0.196152	0.0166374	0.00831872	+	0.999965i	$0.497352\pi$
$140$	0	0
$141$	−5.66025	−0.476679
$142$	0	0
$143$	−4.73205	−0.395714
$144$	0	0
$145$	0.803848	0.0667559
$146$	0	0
$147$	0.732051	0.0603785
$148$	0	0
$149$	−4.73205	−0.387665	−0.193832	−	0.981035i	$-0.562092\pi$
$149$	−4.73205	−0.387665	−0.193832	+	0.981035i	$0.562092\pi$
$150$	0	0
$151$	−8.58846	−0.698919	−0.349459	−	0.936952i	$-0.613635\pi$
$151$	−8.58846	−0.698919	−0.349459	+	0.936952i	$0.613635\pi$
$152$	0	0
$153$	5.41154	0.437497
$154$	0	0
$155$	−2.07180	−0.166411
$156$	0	0
$157$	−10.5885	−0.845051	−0.422525	−	0.906351i	$-0.638856\pi$
$157$	−10.5885	−0.845051	−0.422525	+	0.906351i	$0.638856\pi$
$158$	0	0
$159$	−7.26795	−0.576386
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	−12.3923	−0.970640	−0.485320	−	0.874337i	$-0.661297\pi$
$163$	−12.3923	−0.970640	−0.485320	+	0.874337i	$0.661297\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−16.2679	−1.25885	−0.629426	−	0.777061i	$-0.716710\pi$
$167$	−16.2679	−1.25885	−0.629426	+	0.777061i	$0.716710\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	17.7321	1.35600
$172$	0	0
$173$	−9.12436	−0.693712	−0.346856	−	0.937918i	$-0.612751\pi$
$173$	−9.12436	−0.693712	−0.346856	+	0.937918i	$0.612751\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	7.60770	0.571829
$178$	0	0
$179$	−15.9282	−1.19053	−0.595265	−	0.803530i	$-0.702953\pi$
$179$	−15.9282	−1.19053	−0.595265	+	0.803530i	$0.702953\pi$
$180$	0	0
$181$	4.19615	0.311898	0.155949	−	0.987765i	$-0.450157\pi$
$181$	4.19615	0.311898	0.155949	+	0.987765i	$0.450157\pi$
$182$	0	0
$183$	−7.32051	−0.541148
$184$	0	0
$185$	7.26795	0.534350
$186$	0	0
$187$	10.3923	0.759961
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	12.9282	0.935452	0.467726	−	0.883874i	$-0.345073\pi$
$191$	12.9282	0.935452	0.467726	+	0.883874i	$0.345073\pi$
$192$	0	0
$193$	−2.39230	−0.172202	−0.0861009	−	0.996286i	$-0.527441\pi$
$193$	−2.39230	−0.172202	−0.0861009	+	0.996286i	$0.527441\pi$
$194$	0	0
$195$	1.26795	0.0907997
$196$	0	0
$197$	−12.9282	−0.921096	−0.460548	−	0.887635i	$-0.652347\pi$
$197$	−12.9282	−0.921096	−0.460548	+	0.887635i	$0.652347\pi$
$198$	0	0
$199$	6.19615	0.439234	0.219617	−	0.975586i	$-0.429519\pi$
$199$	6.19615	0.439234	0.219617	+	0.975586i	$0.429519\pi$
$200$	0	0
$201$	10.5359	0.743145
$202$	0	0
$203$	−0.464102	−0.0325735
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	7.39230	0.513801
$208$	0	0
$209$	34.0526	2.35546
$210$	0	0
$211$	−15.3923	−1.05965	−0.529825	−	0.848107i	$-0.677742\pi$
$211$	−15.3923	−1.05965	−0.529825	+	0.848107i	$0.677742\pi$
$212$	0	0
$213$	0.928203	0.0635994
$214$	0	0
$215$	12.1244	0.826874
$216$	0	0
$217$	1.19615	0.0812001
$218$	0	0
$219$	−6.73205	−0.454910
$220$	0	0
$221$	−2.19615	−0.147729
$222$	0	0
$223$	21.1962	1.41940	0.709700	−	0.704504i	$-0.248831\pi$
$223$	21.1962	1.41940	0.709700	+	0.704504i	$0.248831\pi$
$224$	0	0
$225$	4.92820	0.328547
$226$	0	0
$227$	4.39230	0.291528	0.145764	−	0.989319i	$-0.453436\pi$
$227$	4.39230	0.291528	0.145764	+	0.989319i	$0.453436\pi$
$228$	0	0
$229$	16.7846	1.10916	0.554579	−	0.832131i	$-0.312879\pi$
$229$	16.7846	1.10916	0.554579	+	0.832131i	$0.312879\pi$
$230$	0	0
$231$	3.46410	0.227921
$232$	0	0
$233$	9.92820	0.650418	0.325209	−	0.945642i	$-0.394565\pi$
$233$	9.92820	0.650418	0.325209	+	0.945642i	$0.394565\pi$
$234$	0	0
$235$	−13.3923	−0.873618
$236$	0	0
$237$	8.33975	0.541725
$238$	0	0
$239$	4.39230	0.284115	0.142057	−	0.989858i	$-0.454628\pi$
$239$	4.39230	0.284115	0.142057	+	0.989858i	$0.454628\pi$
$240$	0	0
$241$	−7.58846	−0.488816	−0.244408	−	0.969673i	$-0.578594\pi$
$241$	−7.58846	−0.488816	−0.244408	+	0.969673i	$0.578594\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	1.73205	0.110657
$246$	0	0
$247$	−7.19615	−0.457880
$248$	0	0
$249$	0.588457	0.0372920
$250$	0	0
$251$	−19.5167	−1.23188	−0.615940	−	0.787793i	$-0.711224\pi$
$251$	−19.5167	−1.23188	−0.615940	+	0.787793i	$0.711224\pi$
$252$	0	0
$253$	14.1962	0.892504
$254$	0	0
$255$	−2.78461	−0.174379
$256$	0	0
$257$	8.19615	0.511262	0.255631	−	0.966774i	$-0.417717\pi$
$257$	8.19615	0.511262	0.255631	+	0.966774i	$0.417717\pi$
$258$	0	0
$259$	−4.19615	−0.260736
$260$	0	0
$261$	−1.14359	−0.0707867
$262$	0	0
$263$	−9.92820	−0.612199	−0.306100	−	0.951999i	$-0.599024\pi$
$263$	−9.92820	−0.612199	−0.306100	+	0.951999i	$0.599024\pi$
$264$	0	0
$265$	−17.1962	−1.05635
$266$	0	0
$267$	8.19615	0.501596
$268$	0	0
$269$	4.39230	0.267804	0.133902	−	0.990995i	$-0.457249\pi$
$269$	4.39230	0.267804	0.133902	+	0.990995i	$0.457249\pi$
$270$	0	0
$271$	12.7846	0.776610	0.388305	−	0.921531i	$-0.373061\pi$
$271$	12.7846	0.776610	0.388305	+	0.921531i	$0.373061\pi$
$272$	0	0
$273$	−0.732051	−0.0443057
$274$	0	0
$275$	9.46410	0.570707
$276$	0	0
$277$	−23.3923	−1.40551	−0.702754	−	0.711433i	$-0.748046\pi$
$277$	−23.3923	−1.40551	−0.702754	+	0.711433i	$0.748046\pi$
$278$	0	0
$279$	2.94744	0.176459
$280$	0	0
$281$	26.1962	1.56273	0.781366	−	0.624073i	$-0.214523\pi$
$281$	26.1962	1.56273	0.781366	+	0.624073i	$0.214523\pi$
$282$	0	0
$283$	20.3923	1.21220	0.606098	−	0.795390i	$-0.292734\pi$
$283$	20.3923	1.21220	0.606098	+	0.795390i	$0.292734\pi$
$284$	0	0
$285$	−9.12436	−0.540480
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−12.1769	−0.716289
$290$	0	0
$291$	2.05256	0.120323
$292$	0	0
$293$	−4.26795	−0.249336	−0.124668	−	0.992198i	$-0.539787\pi$
$293$	−4.26795	−0.249336	−0.124668	+	0.992198i	$0.539787\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	0	0
$297$	18.9282	1.09833
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−7.00000	−0.403473
$302$	0	0
$303$	11.0718	0.636058
$304$	0	0
$305$	−17.3205	−0.991769
$306$	0	0
$307$	25.5885	1.46041	0.730205	−	0.683228i	$-0.239424\pi$
$307$	25.5885	1.46041	0.730205	+	0.683228i	$0.239424\pi$
$308$	0	0
$309$	−5.85641	−0.333159
$310$	0	0
$311$	−8.87564	−0.503292	−0.251646	−	0.967819i	$-0.580972\pi$
$311$	−8.87564	−0.503292	−0.251646	+	0.967819i	$0.580972\pi$
$312$	0	0
$313$	−19.8038	−1.11938	−0.559690	−	0.828702i	$-0.689080\pi$
$313$	−19.8038	−1.11938	−0.559690	+	0.828702i	$0.689080\pi$
$314$	0	0
$315$	4.26795	0.240472
$316$	0	0
$317$	−25.8564	−1.45224	−0.726120	−	0.687568i	$-0.758678\pi$
$317$	−25.8564	−1.45224	−0.726120	+	0.687568i	$0.758678\pi$
$318$	0	0
$319$	−2.19615	−0.122961
$320$	0	0
$321$	−4.39230	−0.245155
$322$	0	0
$323$	15.8038	0.879350
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−13.3205	−0.736626
$328$	0	0
$329$	7.73205	0.426282
$330$	0	0
$331$	−22.7846	−1.25236	−0.626178	−	0.779680i	$-0.715382\pi$
$331$	−22.7846	−1.25236	−0.626178	+	0.779680i	$0.715382\pi$
$332$	0	0
$333$	−10.3397	−0.566615
$334$	0	0
$335$	24.9282	1.36197
$336$	0	0
$337$	36.1769	1.97068	0.985341	−	0.170596i	$-0.0545693\pi$
$337$	36.1769	1.97068	0.985341	+	0.170596i	$0.0545693\pi$
$338$	0	0
$339$	−13.5167	−0.734124
$340$	0	0
$341$	5.66025	0.306520
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.80385	−0.204792
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	31.1962	1.66989	0.834946	−	0.550332i	$-0.185499\pi$
$349$	31.1962	1.66989	0.834946	+	0.550332i	$0.185499\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−1.85641	−0.0988065	−0.0494033	−	0.998779i	$-0.515732\pi$
$353$	−1.85641	−0.0988065	−0.0494033	+	0.998779i	$0.515732\pi$
$354$	0	0
$355$	2.19615	0.116560
$356$	0	0
$357$	1.60770	0.0850883
$358$	0	0
$359$	26.4449	1.39571	0.697853	−	0.716241i	$-0.254139\pi$
$359$	26.4449	1.39571	0.697853	+	0.716241i	$0.254139\pi$
$360$	0	0
$361$	32.7846	1.72551
$362$	0	0
$363$	8.33975	0.437723
$364$	0	0
$365$	−15.9282	−0.833720
$366$	0	0
$367$	12.1962	0.636634	0.318317	−	0.947984i	$-0.396882\pi$
$367$	12.1962	0.636634	0.318317	+	0.947984i	$0.396882\pi$
$368$	0	0
$369$	−8.53590	−0.444361
$370$	0	0
$371$	9.92820	0.515447
$372$	0	0
$373$	−24.7846	−1.28330	−0.641649	−	0.766998i	$-0.721749\pi$
$373$	−24.7846	−1.28330	−0.641649	+	0.766998i	$0.721749\pi$
$374$	0	0
$375$	−8.87564	−0.458336
$376$	0	0
$377$	0.464102	0.0239024
$378$	0	0
$379$	1.80385	0.0926574	0.0463287	−	0.998926i	$-0.485248\pi$
$379$	1.80385	0.0926574	0.0463287	+	0.998926i	$0.485248\pi$
$380$	0	0
$381$	10.5359	0.539770
$382$	0	0
$383$	11.3205	0.578451	0.289225	−	0.957261i	$-0.406602\pi$
$383$	11.3205	0.578451	0.289225	+	0.957261i	$0.406602\pi$
$384$	0	0
$385$	8.19615	0.417715
$386$	0	0
$387$	−17.2487	−0.876801
$388$	0	0
$389$	−14.5359	−0.736999	−0.368500	−	0.929628i	$-0.620128\pi$
$389$	−14.5359	−0.736999	−0.368500	+	0.929628i	$0.620128\pi$
$390$	0	0
$391$	6.58846	0.333193
$392$	0	0
$393$	−10.6410	−0.536768
$394$	0	0
$395$	19.7321	0.992827
$396$	0	0
$397$	11.5885	0.581608	0.290804	−	0.956783i	$-0.406077\pi$
$397$	11.5885	0.581608	0.290804	+	0.956783i	$0.406077\pi$
$398$	0	0
$399$	5.26795	0.263727
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	−1.19615	−0.0595846
$404$	0	0
$405$	7.73205	0.384209
$406$	0	0
$407$	−19.8564	−0.984246
$408$	0	0
$409$	−1.58846	−0.0785442	−0.0392721	−	0.999229i	$-0.512504\pi$
$409$	−1.58846	−0.0785442	−0.0392721	+	0.999229i	$0.512504\pi$
$410$	0	0
$411$	9.21539	0.454562
$412$	0	0
$413$	−10.3923	−0.511372
$414$	0	0
$415$	1.39230	0.0683456
$416$	0	0
$417$	0.143594	0.00703181
$418$	0	0
$419$	31.5167	1.53969	0.769845	−	0.638231i	$-0.220334\pi$
$419$	31.5167	1.53969	0.769845	+	0.638231i	$0.220334\pi$
$420$	0	0
$421$	20.5885	1.00342	0.501710	−	0.865036i	$-0.332704\pi$
$421$	20.5885	1.00342	0.501710	+	0.865036i	$0.332704\pi$
$422$	0	0
$423$	19.0526	0.926367
$424$	0	0
$425$	4.39230	0.213058
$426$	0	0
$427$	10.0000	0.483934
$428$	0	0
$429$	−3.46410	−0.167248
$430$	0	0
$431$	−24.9282	−1.20075	−0.600375	−	0.799719i	$-0.704982\pi$
$431$	−24.9282	−1.20075	−0.600375	+	0.799719i	$0.704982\pi$
$432$	0	0
$433$	4.78461	0.229934	0.114967	−	0.993369i	$-0.463324\pi$
$433$	4.78461	0.229934	0.114967	+	0.993369i	$0.463324\pi$
$434$	0	0
$435$	0.588457	0.0282144
$436$	0	0
$437$	21.5885	1.03272
$438$	0	0
$439$	−6.39230	−0.305088	−0.152544	−	0.988297i	$-0.548747\pi$
$439$	−6.39230	−0.305088	−0.152544	+	0.988297i	$0.548747\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	6.46410	0.307119	0.153559	−	0.988139i	$-0.450926\pi$
$443$	6.46410	0.307119	0.153559	+	0.988139i	$0.450926\pi$
$444$	0	0
$445$	19.3923	0.919283
$446$	0	0
$447$	−3.46410	−0.163846
$448$	0	0
$449$	23.3205	1.10056	0.550281	−	0.834979i	$-0.314520\pi$
$449$	23.3205	1.10056	0.550281	+	0.834979i	$0.314520\pi$
$450$	0	0
$451$	−16.3923	−0.771883
$452$	0	0
$453$	−6.28719	−0.295398
$454$	0	0
$455$	−1.73205	−0.0811998
$456$	0	0
$457$	20.5885	0.963087	0.481544	−	0.876422i	$-0.340076\pi$
$457$	20.5885	0.963087	0.481544	+	0.876422i	$0.340076\pi$
$458$	0	0
$459$	8.78461	0.410030
$460$	0	0
$461$	−39.4641	−1.83803	−0.919013	−	0.394227i	$-0.871012\pi$
$461$	−39.4641	−1.83803	−0.919013	+	0.394227i	$0.871012\pi$
$462$	0	0
$463$	10.5885	0.492087	0.246044	−	0.969259i	$-0.420869\pi$
$463$	10.5885	0.492087	0.246044	+	0.969259i	$0.420869\pi$
$464$	0	0
$465$	−1.51666	−0.0703334
$466$	0	0
$467$	−35.9090	−1.66167	−0.830834	−	0.556520i	$-0.812136\pi$
$467$	−35.9090	−1.66167	−0.830834	+	0.556520i	$0.812136\pi$
$468$	0	0
$469$	−14.3923	−0.664575
$470$	0	0
$471$	−7.75129	−0.357161
$472$	0	0
$473$	−33.1244	−1.52306
$474$	0	0
$475$	14.3923	0.660364
$476$	0	0
$477$	24.4641	1.12013
$478$	0	0
$479$	−31.7321	−1.44987	−0.724937	−	0.688815i	$-0.758131\pi$
$479$	−31.7321	−1.44987	−0.724937	+	0.688815i	$0.758131\pi$
$480$	0	0
$481$	4.19615	0.191328
$482$	0	0
$483$	2.19615	0.0999284
$484$	0	0
$485$	4.85641	0.220518
$486$	0	0
$487$	2.39230	0.108406	0.0542028	−	0.998530i	$-0.482738\pi$
$487$	2.39230	0.108406	0.0542028	+	0.998530i	$0.482738\pi$
$488$	0	0
$489$	−9.07180	−0.410241
$490$	0	0
$491$	−24.2487	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$491$	−24.2487	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$492$	0	0
$493$	−1.01924	−0.0459042
$494$	0	0
$495$	20.1962	0.907750
$496$	0	0
$497$	−1.26795	−0.0568753
$498$	0	0
$499$	−28.1962	−1.26223	−0.631117	−	0.775688i	$-0.717403\pi$
$499$	−28.1962	−1.26223	−0.631117	+	0.775688i	$0.717403\pi$
$500$	0	0
$501$	−11.9090	−0.532053
$502$	0	0
$503$	−33.7128	−1.50318	−0.751590	−	0.659631i	$-0.770713\pi$
$503$	−33.7128	−1.50318	−0.751590	+	0.659631i	$0.770713\pi$
$504$	0	0
$505$	26.1962	1.16571
$506$	0	0
$507$	0.732051	0.0325115
$508$	0	0
$509$	−39.5885	−1.75473	−0.877364	−	0.479826i	$-0.840700\pi$
$509$	−39.5885	−1.75473	−0.877364	+	0.479826i	$0.840700\pi$
$510$	0	0
$511$	9.19615	0.406814
$512$	0	0
$513$	28.7846	1.27087
$514$	0	0
$515$	−13.8564	−0.610586
$516$	0	0
$517$	36.5885	1.60916
$518$	0	0
$519$	−6.67949	−0.293197
$520$	0	0
$521$	17.0718	0.747929	0.373964	−	0.927443i	$-0.377998\pi$
$521$	17.0718	0.747929	0.373964	+	0.927443i	$0.377998\pi$
$522$	0	0
$523$	−10.7846	−0.471578	−0.235789	−	0.971804i	$-0.575767\pi$
$523$	−10.7846	−0.471578	−0.235789	+	0.971804i	$0.575767\pi$
$524$	0	0
$525$	1.46410	0.0638986
$526$	0	0
$527$	2.62693	0.114431
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−25.6077	−1.11128
$532$	0	0
$533$	3.46410	0.150047
$534$	0	0
$535$	−10.3923	−0.449299
$536$	0	0
$537$	−11.6603	−0.503177
$538$	0	0
$539$	−4.73205	−0.203824
$540$	0	0
$541$	35.8038	1.53933	0.769664	−	0.638449i	$-0.220424\pi$
$541$	35.8038	1.53933	0.769664	+	0.638449i	$0.220424\pi$
$542$	0	0
$543$	3.07180	0.131823
$544$	0	0
$545$	−31.5167	−1.35003
$546$	0	0
$547$	15.7846	0.674901	0.337451	−	0.941343i	$-0.390435\pi$
$547$	15.7846	0.674901	0.337451	+	0.941343i	$0.390435\pi$
$548$	0	0
$549$	24.6410	1.05165
$550$	0	0
$551$	−3.33975	−0.142278
$552$	0	0
$553$	−11.3923	−0.484450
$554$	0	0
$555$	5.32051	0.225843
$556$	0	0
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	0	0
$559$	7.00000	0.296068
$560$	0	0
$561$	7.60770	0.321197
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−31.9808	−1.34544
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	−37.3923	−1.56757	−0.783783	−	0.621034i	$-0.786713\pi$
$569$	−37.3923	−1.56757	−0.783783	+	0.621034i	$0.786713\pi$
$570$	0	0
$571$	−2.21539	−0.0927112	−0.0463556	−	0.998925i	$-0.514761\pi$
$571$	−2.21539	−0.0927112	−0.0463556	+	0.998925i	$0.514761\pi$
$572$	0	0
$573$	9.46410	0.395369
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	−1.75129	−0.0727811
$580$	0	0
$581$	−0.803848	−0.0333492
$582$	0	0
$583$	46.9808	1.94574
$584$	0	0
$585$	−4.26795	−0.176458
$586$	0	0
$587$	24.1244	0.995719	0.497859	−	0.867258i	$-0.334120\pi$
$587$	24.1244	0.995719	0.497859	+	0.867258i	$0.334120\pi$
$588$	0	0
$589$	8.60770	0.354674
$590$	0	0
$591$	−9.46410	−0.389301
$592$	0	0
$593$	21.3397	0.876318	0.438159	−	0.898897i	$-0.355631\pi$
$593$	21.3397	0.876318	0.438159	+	0.898897i	$0.355631\pi$
$594$	0	0
$595$	3.80385	0.155943
$596$	0	0
$597$	4.53590	0.185642
$598$	0	0
$599$	−42.4641	−1.73504	−0.867518	−	0.497406i	$-0.834286\pi$
$599$	−42.4641	−1.73504	−0.867518	+	0.497406i	$0.834286\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−35.4641	−1.44421
$604$	0	0
$605$	19.7321	0.802222
$606$	0	0
$607$	−20.5885	−0.835660	−0.417830	−	0.908525i	$-0.637209\pi$
$607$	−20.5885	−0.835660	−0.417830	+	0.908525i	$0.637209\pi$
$608$	0	0
$609$	−0.339746	−0.0137672
$610$	0	0
$611$	−7.73205	−0.312805
$612$	0	0
$613$	−0.784610	−0.0316901	−0.0158450	−	0.999874i	$-0.505044\pi$
$613$	−0.784610	−0.0316901	−0.0158450	+	0.999874i	$0.505044\pi$
$614$	0	0
$615$	4.39230	0.177115
$616$	0	0
$617$	21.7128	0.874125	0.437062	−	0.899431i	$-0.356019\pi$
$617$	21.7128	0.874125	0.437062	+	0.899431i	$0.356019\pi$
$618$	0	0
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	12.0000	0.481543
$622$	0	0
$623$	−11.1962	−0.448564
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	24.9282	0.995537
$628$	0	0
$629$	−9.21539	−0.367442
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	0	0
$633$	−11.2679	−0.447861
$634$	0	0
$635$	24.9282	0.989246
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−3.12436	−0.123598
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	−0.392305	−0.0154710	−0.00773550	−	0.999970i	$-0.502462\pi$
$643$	−0.392305	−0.0154710	−0.00773550	+	0.999970i	$0.502462\pi$
$644$	0	0
$645$	8.87564	0.349478
$646$	0	0
$647$	10.0526	0.395207	0.197603	−	0.980282i	$-0.436684\pi$
$647$	10.0526	0.395207	0.197603	+	0.980282i	$0.436684\pi$
$648$	0	0
$649$	−49.1769	−1.93036
$650$	0	0
$651$	0.875644	0.0343192
$652$	0	0
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	−25.1769	−0.983743
$656$	0	0
$657$	22.6603	0.884061
$658$	0	0
$659$	−23.5359	−0.916828	−0.458414	−	0.888739i	$-0.651582\pi$
$659$	−23.5359	−0.916828	−0.458414	+	0.888739i	$0.651582\pi$
$660$	0	0
$661$	−47.9808	−1.86624	−0.933118	−	0.359571i	$-0.882923\pi$
$661$	−47.9808	−1.86624	−0.933118	+	0.359571i	$0.882923\pi$
$662$	0	0
$663$	−1.60770	−0.0624377
$664$	0	0
$665$	12.4641	0.483337
$666$	0	0
$667$	−1.39230	−0.0539103
$668$	0	0
$669$	15.5167	0.599909
$670$	0	0
$671$	47.3205	1.82679
$672$	0	0
$673$	3.39230	0.130764	0.0653819	−	0.997860i	$-0.479173\pi$
$673$	3.39230	0.130764	0.0653819	+	0.997860i	$0.479173\pi$
$674$	0	0
$675$	8.00000	0.307920
$676$	0	0
$677$	−29.6603	−1.13994	−0.569968	−	0.821667i	$-0.693044\pi$
$677$	−29.6603	−1.13994	−0.569968	+	0.821667i	$0.693044\pi$
$678$	0	0
$679$	−2.80385	−0.107602
$680$	0	0
$681$	3.21539	0.123214
$682$	0	0
$683$	16.3923	0.627234	0.313617	−	0.949550i	$-0.398459\pi$
$683$	16.3923	0.627234	0.313617	+	0.949550i	$0.398459\pi$
$684$	0	0
$685$	21.8038	0.833082
$686$	0	0
$687$	12.2872	0.468785
$688$	0	0
$689$	−9.92820	−0.378234
$690$	0	0
$691$	7.58846	0.288679	0.144339	−	0.989528i	$-0.453894\pi$
$691$	7.58846	0.288679	0.144339	+	0.989528i	$0.453894\pi$
$692$	0	0
$693$	−11.6603	−0.442936
$694$	0	0
$695$	0.339746	0.0128873
$696$	0	0
$697$	−7.60770	−0.288162
$698$	0	0
$699$	7.26795	0.274899
$700$	0	0
$701$	−40.1769	−1.51746	−0.758731	−	0.651405i	$-0.774180\pi$
$701$	−40.1769	−1.51746	−0.758731	+	0.651405i	$0.774180\pi$
$702$	0	0
$703$	−30.1962	−1.13887
$704$	0	0
$705$	−9.80385	−0.369234
$706$	0	0
$707$	−15.1244	−0.568810
$708$	0	0
$709$	14.5885	0.547881	0.273941	−	0.961747i	$-0.411673\pi$
$709$	14.5885	0.547881	0.273941	+	0.961747i	$0.411673\pi$
$710$	0	0
$711$	−28.0718	−1.05277
$712$	0	0
$713$	3.58846	0.134389
$714$	0	0
$715$	−8.19615	−0.306519
$716$	0	0
$717$	3.21539	0.120081
$718$	0	0
$719$	−17.6603	−0.658616	−0.329308	−	0.944222i	$-0.606815\pi$
$719$	−17.6603	−0.658616	−0.329308	+	0.944222i	$0.606815\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−5.55514	−0.206598
$724$	0	0
$725$	−0.928203	−0.0344726
$726$	0	0
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−15.3731	−0.568593
$732$	0	0
$733$	29.5885	1.09287	0.546437	−	0.837500i	$-0.315984\pi$
$733$	29.5885	1.09287	0.546437	+	0.837500i	$0.315984\pi$
$734$	0	0
$735$	1.26795	0.0467690
$736$	0	0
$737$	−68.1051	−2.50868
$738$	0	0
$739$	−47.3731	−1.74265	−0.871323	−	0.490710i	$-0.836738\pi$
$739$	−47.3731	−1.74265	−0.871323	+	0.490710i	$0.836738\pi$
$740$	0	0
$741$	−5.26795	−0.193523
$742$	0	0
$743$	−34.6410	−1.27086	−0.635428	−	0.772160i	$-0.719176\pi$
$743$	−34.6410	−1.27086	−0.635428	+	0.772160i	$0.719176\pi$
$744$	0	0
$745$	−8.19615	−0.300284
$746$	0	0
$747$	−1.98076	−0.0724723
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	35.3923	1.29148	0.645742	−	0.763556i	$-0.276548\pi$
$751$	35.3923	1.29148	0.645742	+	0.763556i	$0.276548\pi$
$752$	0	0
$753$	−14.2872	−0.520654
$754$	0	0
$755$	−14.8756	−0.541380
$756$	0	0
$757$	12.6077	0.458234	0.229117	−	0.973399i	$-0.426416\pi$
$757$	12.6077	0.458234	0.229117	+	0.973399i	$0.426416\pi$
$758$	0	0
$759$	10.3923	0.377217
$760$	0	0
$761$	25.0526	0.908155	0.454077	−	0.890962i	$-0.349969\pi$
$761$	25.0526	0.908155	0.454077	+	0.890962i	$0.349969\pi$
$762$	0	0
$763$	18.1962	0.658745
$764$	0	0
$765$	9.37307	0.338884
$766$	0	0
$767$	10.3923	0.375244
$768$	0	0
$769$	32.3731	1.16740	0.583701	−	0.811968i	$-0.301604\pi$
$769$	32.3731	1.16740	0.583701	+	0.811968i	$0.301604\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−13.6077	−0.489435	−0.244717	−	0.969594i	$-0.578695\pi$
$773$	−13.6077	−0.489435	−0.244717	+	0.969594i	$0.578695\pi$
$774$	0	0
$775$	2.39230	0.0859341
$776$	0	0
$777$	−3.07180	−0.110200
$778$	0	0
$779$	−24.9282	−0.893146
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	−1.85641	−0.0663426
$784$	0	0
$785$	−18.3397	−0.654574
$786$	0	0
$787$	−9.98076	−0.355776	−0.177888	−	0.984051i	$-0.556926\pi$
$787$	−9.98076	−0.355776	−0.177888	+	0.984051i	$0.556926\pi$
$788$	0	0
$789$	−7.26795	−0.258746
$790$	0	0
$791$	18.4641	0.656508
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	−12.5885	−0.446467
$796$	0	0
$797$	−40.6410	−1.43958	−0.719789	−	0.694193i	$-0.755762\pi$
$797$	−40.6410	−1.43958	−0.719789	+	0.694193i	$0.755762\pi$
$798$	0	0
$799$	16.9808	0.600736
$800$	0	0
$801$	−27.5885	−0.974790
$802$	0	0
$803$	43.5167	1.53567
$804$	0	0
$805$	5.19615	0.183140
$806$	0	0
$807$	3.21539	0.113187
$808$	0	0
$809$	−48.4641	−1.70391	−0.851954	−	0.523617i	$-0.824582\pi$
$809$	−48.4641	−1.70391	−0.851954	+	0.523617i	$0.824582\pi$
$810$	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
$813$	9.35898	0.328234
$814$	0	0
$815$	−21.4641	−0.751855
$816$	0	0
$817$	−50.3731	−1.76233
$818$	0	0
$819$	2.46410	0.0861027
$820$	0	0
$821$	10.3923	0.362694	0.181347	−	0.983419i	$-0.441954\pi$
$821$	10.3923	0.362694	0.181347	+	0.983419i	$0.441954\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	0	0
$825$	6.92820	0.241209
$826$	0	0
$827$	−28.3923	−0.987297	−0.493649	−	0.869661i	$-0.664337\pi$
$827$	−28.3923	−0.987297	−0.493649	+	0.869661i	$0.664337\pi$
$828$	0	0
$829$	20.5885	0.715067	0.357533	−	0.933900i	$-0.383618\pi$
$829$	20.5885	0.715067	0.357533	+	0.933900i	$0.383618\pi$
$830$	0	0
$831$	−17.1244	−0.594037
$832$	0	0
$833$	−2.19615	−0.0760922
$834$	0	0
$835$	−28.1769	−0.975102
$836$	0	0
$837$	4.78461	0.165380
$838$	0	0
$839$	21.4641	0.741023	0.370512	−	0.928828i	$-0.379182\pi$
$839$	21.4641	0.741023	0.370512	+	0.928828i	$0.379182\pi$
$840$	0	0
$841$	−28.7846	−0.992573
$842$	0	0
$843$	19.1769	0.660488
$844$	0	0
$845$	1.73205	0.0595844
$846$	0	0
$847$	−11.3923	−0.391444
$848$	0	0
$849$	14.9282	0.512335
$850$	0	0
$851$	−12.5885	−0.431527
$852$	0	0
$853$	−37.5885	−1.28700	−0.643502	−	0.765444i	$-0.722519\pi$
$853$	−37.5885	−1.28700	−0.643502	+	0.765444i	$0.722519\pi$
$854$	0	0
$855$	30.7128	1.05036
$856$	0	0
$857$	−17.3205	−0.591657	−0.295829	−	0.955241i	$-0.595596\pi$
$857$	−17.3205	−0.591657	−0.295829	+	0.955241i	$0.595596\pi$
$858$	0	0
$859$	42.7846	1.45979	0.729896	−	0.683558i	$-0.239568\pi$
$859$	42.7846	1.45979	0.729896	+	0.683558i	$0.239568\pi$
$860$	0	0
$861$	−2.53590	−0.0864232
$862$	0	0
$863$	−3.71281	−0.126386	−0.0631928	−	0.998001i	$-0.520128\pi$
$863$	−3.71281	−0.126386	−0.0631928	+	0.998001i	$0.520128\pi$
$864$	0	0
$865$	−15.8038	−0.537347
$866$	0	0
$867$	−8.91412	−0.302739
$868$	0	0
$869$	−53.9090	−1.82874
$870$	0	0
$871$	14.3923	0.487665
$872$	0	0
$873$	−6.90897	−0.233833
$874$	0	0
$875$	12.1244	0.409878
$876$	0	0
$877$	11.8038	0.398588	0.199294	−	0.979940i	$-0.436135\pi$
$877$	11.8038	0.398588	0.199294	+	0.979940i	$0.436135\pi$
$878$	0	0
$879$	−3.12436	−0.105382
$880$	0	0
$881$	15.4641	0.520999	0.260499	−	0.965474i	$-0.416113\pi$
$881$	15.4641	0.520999	0.260499	+	0.965474i	$0.416113\pi$
$882$	0	0
$883$	−51.1769	−1.72224	−0.861120	−	0.508402i	$-0.830237\pi$
$883$	−51.1769	−1.72224	−0.861120	+	0.508402i	$0.830237\pi$
$884$	0	0
$885$	13.1769	0.442937
$886$	0	0
$887$	16.3923	0.550400	0.275200	−	0.961387i	$-0.411256\pi$
$887$	16.3923	0.550400	0.275200	+	0.961387i	$0.411256\pi$
$888$	0	0
$889$	−14.3923	−0.482702
$890$	0	0
$891$	−21.1244	−0.707693
$892$	0	0
$893$	55.6410	1.86196
$894$	0	0
$895$	−27.5885	−0.922180
$896$	0	0
$897$	−2.19615	−0.0733274
$898$	0	0
$899$	−0.555136	−0.0185148
$900$	0	0
$901$	21.8038	0.726391
$902$	0	0
$903$	−5.12436	−0.170528
$904$	0	0
$905$	7.26795	0.241595
$906$	0	0
$907$	50.1769	1.66610	0.833049	−	0.553200i	$-0.186593\pi$
$907$	50.1769	1.66610	0.833049	+	0.553200i	$0.186593\pi$
$908$	0	0
$909$	−37.2679	−1.23610
$910$	0	0
$911$	38.5692	1.27786	0.638928	−	0.769267i	$-0.279378\pi$
$911$	38.5692	1.27786	0.638928	+	0.769267i	$0.279378\pi$
$912$	0	0
$913$	−3.80385	−0.125889
$914$	0	0
$915$	−12.6795	−0.419171
$916$	0	0
$917$	14.5359	0.480018
$918$	0	0
$919$	−40.7846	−1.34536	−0.672680	−	0.739933i	$-0.734857\pi$
$919$	−40.7846	−1.34536	−0.672680	+	0.739933i	$0.734857\pi$
$920$	0	0
$921$	18.7321	0.617242
$922$	0	0
$923$	1.26795	0.0417351
$924$	0	0
$925$	−8.39230	−0.275937
$926$	0	0
$927$	19.7128	0.647454
$928$	0	0
$929$	−0.124356	−0.00407998	−0.00203999	−	0.999998i	$-0.500649\pi$
$929$	−0.124356	−0.00407998	−0.00203999	+	0.999998i	$0.500649\pi$
$930$	0	0
$931$	−7.19615	−0.235844
$932$	0	0
$933$	−6.49742	−0.212716
$934$	0	0
$935$	18.0000	0.588663
$936$	0	0
$937$	58.1962	1.90119	0.950593	−	0.310441i	$-0.100477\pi$
$937$	58.1962	1.90119	0.950593	+	0.310441i	$0.100477\pi$
$938$	0	0
$939$	−14.4974	−0.473106
$940$	0	0
$941$	−46.5167	−1.51640	−0.758200	−	0.652022i	$-0.773921\pi$
$941$	−46.5167	−1.51640	−0.758200	+	0.652022i	$0.773921\pi$
$942$	0	0
$943$	−10.3923	−0.338420
$944$	0	0
$945$	6.92820	0.225374
$946$	0	0
$947$	25.6077	0.832138	0.416069	−	0.909333i	$-0.363407\pi$
$947$	25.6077	0.832138	0.416069	+	0.909333i	$0.363407\pi$
$948$	0	0
$949$	−9.19615	−0.298520
$950$	0	0
$951$	−18.9282	−0.613789
$952$	0	0
$953$	−45.2487	−1.46575	−0.732875	−	0.680364i	$-0.761822\pi$
$953$	−45.2487	−1.46575	−0.732875	+	0.680364i	$0.761822\pi$
$954$	0	0
$955$	22.3923	0.724598
$956$	0	0
$957$	−1.60770	−0.0519694
$958$	0	0
$959$	−12.5885	−0.406502
$960$	0	0
$961$	−29.5692	−0.953846
$962$	0	0
$963$	14.7846	0.476427
$964$	0	0
$965$	−4.14359	−0.133387
$966$	0	0
$967$	−34.1962	−1.09967	−0.549837	−	0.835272i	$-0.685310\pi$
$967$	−34.1962	−1.09967	−0.549837	+	0.835272i	$0.685310\pi$
$968$	0	0
$969$	11.5692	0.371657
$970$	0	0
$971$	−39.4641	−1.26646	−0.633232	−	0.773962i	$-0.718272\pi$
$971$	−39.4641	−1.26646	−0.633232	+	0.773962i	$0.718272\pi$
$972$	0	0
$973$	−0.196152	−0.00628836
$974$	0	0
$975$	−1.46410	−0.0468888
$976$	0	0
$977$	16.7321	0.535306	0.267653	−	0.963515i	$-0.413752\pi$
$977$	16.7321	0.535306	0.267653	+	0.963515i	$0.413752\pi$
$978$	0	0
$979$	−52.9808	−1.69327
$980$	0	0
$981$	44.8372	1.43154
$982$	0	0
$983$	−8.66025	−0.276219	−0.138110	−	0.990417i	$-0.544103\pi$
$983$	−8.66025	−0.276219	−0.138110	+	0.990417i	$0.544103\pi$
$984$	0	0
$985$	−22.3923	−0.713478
$986$	0	0
$987$	5.66025	0.180168
$988$	0	0
$989$	−21.0000	−0.667761
$990$	0	0
$991$	−9.60770	−0.305198	−0.152599	−	0.988288i	$-0.548764\pi$
$991$	−9.60770	−0.305198	−0.152599	+	0.988288i	$0.548764\pi$
$992$	0	0
$993$	−16.6795	−0.529308
$994$	0	0
$995$	10.7321	0.340229
$996$	0	0
$997$	57.1769	1.81081	0.905406	−	0.424548i	$-0.139567\pi$
$997$	57.1769	1.81081	0.905406	+	0.424548i	$0.139567\pi$
$998$	0	0
$999$	−16.7846	−0.531042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.n.1.2		2
4.3	odd	2		364.2.a.d.1.1	✓	2
8.3	odd	2		5824.2.a.bh.1.2		2
8.5	even	2		5824.2.a.bq.1.1		2
12.11	even	2		3276.2.a.n.1.1		2
20.19	odd	2		9100.2.a.o.1.2		2
28.3	even	6		2548.2.j.n.1353.1		4
28.11	odd	6		2548.2.j.k.1353.2		4
28.19	even	6		2548.2.j.n.1145.1		4
28.23	odd	6		2548.2.j.k.1145.2		4
28.27	even	2		2548.2.a.l.1.2		2
52.31	even	4		4732.2.g.h.337.1		4
52.47	even	4		4732.2.g.h.337.2		4
52.51	odd	2		4732.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.a.d.1.1	✓	2	4.3	odd	2
1456.2.a.n.1.2		2	1.1	even	1	trivial
2548.2.a.l.1.2		2	28.27	even	2
2548.2.j.k.1145.2		4	28.23	odd	6
2548.2.j.k.1353.2		4	28.11	odd	6
2548.2.j.n.1145.1		4	28.19	even	6
2548.2.j.n.1353.1		4	28.3	even	6
3276.2.a.n.1.1		2	12.11	even	2
4732.2.a.l.1.1		2	52.51	odd	2
4732.2.g.h.337.1		4	52.31	even	4
4732.2.g.h.337.2		4	52.47	even	4
5824.2.a.bh.1.2		2	8.3	odd	2
5824.2.a.bq.1.1		2	8.5	even	2
9100.2.a.o.1.2		2	20.19	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)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} +1.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} +1.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} -4.73205 q^{11} +1.00000 q^{13} +1.26795 q^{15} -2.19615 q^{17} -7.19615 q^{19} -0.732051 q^{21} -3.00000 q^{23} -2.00000 q^{25} -4.00000 q^{27} +0.464102 q^{29} -1.19615 q^{31} -3.46410 q^{33} -1.73205 q^{35} +4.19615 q^{37} +0.732051 q^{39} +3.46410 q^{41} +7.00000 q^{43} -4.26795 q^{45} -7.73205 q^{47} +1.00000 q^{49} -1.60770 q^{51} -9.92820 q^{53} -8.19615 q^{55} -5.26795 q^{57} +10.3923 q^{59} -10.0000 q^{61} +2.46410 q^{63} +1.73205 q^{65} +14.3923 q^{67} -2.19615 q^{69} +1.26795 q^{71} -9.19615 q^{73} -1.46410 q^{75} +4.73205 q^{77} +11.3923 q^{79} +4.46410 q^{81} +0.803848 q^{83} -3.80385 q^{85} +0.339746 q^{87} +11.1962 q^{89} -1.00000 q^{91} -0.875644 q^{93} -12.4641 q^{95} +2.80385 q^{97} +11.6603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 6 q^{23} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 8 q^{31} - 2 q^{37} - 2 q^{39} + 14 q^{43} - 12 q^{45} - 12 q^{47} + 2 q^{49} - 24 q^{51} - 6 q^{53} - 6 q^{55} - 14 q^{57} - 20 q^{61} - 2 q^{63} + 8 q^{67} + 6 q^{69} + 6 q^{71} - 8 q^{73} + 4 q^{75} + 6 q^{77} + 2 q^{79} + 2 q^{81} + 12 q^{83} - 18 q^{85} + 18 q^{87} + 12 q^{89} - 2 q^{91} - 26 q^{93} - 18 q^{95} + 16 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 - 6 * q^23 - 4 * q^25 - 8 * q^27 - 6 * q^29 + 8 * q^31 - 2 * q^37 - 2 * q^39 + 14 * q^43 - 12 * q^45 - 12 * q^47 + 2 * q^49 - 24 * q^51 - 6 * q^53 - 6 * q^55 - 14 * q^57 - 20 * q^61 - 2 * q^63 + 8 * q^67 + 6 * q^69 + 6 * q^71 - 8 * q^73 + 4 * q^75 + 6 * q^77 + 2 * q^79 + 2 * q^81 + 12 * q^83 - 18 * q^85 + 18 * q^87 + 12 * q^89 - 2 * q^91 - 26 * q^93 - 18 * q^95 + 16 * q^97 + 6 * q^99

Introduction

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