LMFDB - Embedded newform 1456.2.a.r.1.1

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			1.1
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)

$f(q)$	$=$	$q-0.732051 q^{3} -1.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} -1.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} -0.732051 q^{11} -1.00000 q^{13} +1.26795 q^{15} -0.732051 q^{17} +1.73205 q^{19} -0.732051 q^{21} +6.46410 q^{23} -2.00000 q^{25} +4.00000 q^{27} +0.464102 q^{29} +3.73205 q^{31} +0.535898 q^{33} -1.73205 q^{35} -6.73205 q^{37} +0.732051 q^{39} +10.3923 q^{41} +5.53590 q^{43} +4.26795 q^{45} +2.26795 q^{47} +1.00000 q^{49} +0.535898 q^{51} +3.92820 q^{53} +1.26795 q^{55} -1.26795 q^{57} +3.46410 q^{59} -2.00000 q^{61} -2.46410 q^{63} +1.73205 q^{65} -10.3923 q^{67} -4.73205 q^{69} +0.196152 q^{71} +6.26795 q^{73} +1.46410 q^{75} -0.732051 q^{77} +9.92820 q^{79} +4.46410 q^{81} +9.73205 q^{83} +1.26795 q^{85} -0.339746 q^{87} +15.7321 q^{89} -1.00000 q^{91} -2.73205 q^{93} -3.00000 q^{95} +8.12436 q^{97} +1.80385 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−0.732051	−0.220722	−0.110361	−	0.993892i	$-0.535201\pi$
$11$	−0.732051	−0.220722	−0.110361	+	0.993892i	$0.535201\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.26795	0.327383
$16$	0	0
$17$	−0.732051	−0.177548	−0.0887742	−	0.996052i	$-0.528295\pi$
$17$	−0.732051	−0.177548	−0.0887742	+	0.996052i	$0.528295\pi$
$18$	0	0
$19$	1.73205	0.397360	0.198680	−	0.980064i	$-0.436335\pi$
$19$	1.73205	0.397360	0.198680	+	0.980064i	$0.436335\pi$
$20$	0	0
$21$	−0.732051	−0.159747
$22$	0	0
$23$	6.46410	1.34786	0.673929	−	0.738796i	$-0.264605\pi$
$23$	6.46410	1.34786	0.673929	+	0.738796i	$0.264605\pi$
$24$	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
$31$	3.73205	0.670296	0.335148	−	0.942165i	$-0.391214\pi$
$31$	3.73205	0.670296	0.335148	+	0.942165i	$0.391214\pi$
$32$	0	0
$33$	0.535898	0.0932879
$34$	0	0
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−6.73205	−1.10674	−0.553371	−	0.832935i	$-0.686659\pi$
$37$	−6.73205	−1.10674	−0.553371	+	0.832935i	$0.686659\pi$
$38$	0	0
$39$	0.732051	0.117222
$40$	0	0
$41$	10.3923	1.62301	0.811503	−	0.584349i	$-0.198650\pi$
$41$	10.3923	1.62301	0.811503	+	0.584349i	$0.198650\pi$
$42$	0	0
$43$	5.53590	0.844217	0.422108	−	0.906545i	$-0.361290\pi$
$43$	5.53590	0.844217	0.422108	+	0.906545i	$0.361290\pi$
$44$	0	0
$45$	4.26795	0.636228
$46$	0	0
$47$	2.26795	0.330814	0.165407	−	0.986225i	$-0.447106\pi$
$47$	2.26795	0.330814	0.165407	+	0.986225i	$0.447106\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.535898	0.0750408
$52$	0	0
$53$	3.92820	0.539580	0.269790	−	0.962919i	$-0.413046\pi$
$53$	3.92820	0.539580	0.269790	+	0.962919i	$0.413046\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	0	0
$57$	−1.26795	−0.167944
$58$	0	0
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	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
$63$	−2.46410	−0.310448
$64$	0	0
$65$	1.73205	0.214834
$66$	0	0
$67$	−10.3923	−1.26962	−0.634811	−	0.772667i	$-0.718922\pi$
$67$	−10.3923	−1.26962	−0.634811	+	0.772667i	$0.718922\pi$
$68$	0	0
$69$	−4.73205	−0.569672
$70$	0	0
$71$	0.196152	0.0232790	0.0116395	−	0.999932i	$-0.496295\pi$
$71$	0.196152	0.0232790	0.0116395	+	0.999932i	$0.496295\pi$
$72$	0	0
$73$	6.26795	0.733608	0.366804	−	0.930298i	$-0.380452\pi$
$73$	6.26795	0.733608	0.366804	+	0.930298i	$0.380452\pi$
$74$	0	0
$75$	1.46410	0.169060
$76$	0	0
$77$	−0.732051	−0.0834249
$78$	0	0
$79$	9.92820	1.11701	0.558505	−	0.829501i	$-0.311375\pi$
$79$	9.92820	1.11701	0.558505	+	0.829501i	$0.311375\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	9.73205	1.06823	0.534116	−	0.845411i	$-0.320645\pi$
$83$	9.73205	1.06823	0.534116	+	0.845411i	$0.320645\pi$
$84$	0	0
$85$	1.26795	0.137528
$86$	0	0
$87$	−0.339746	−0.0364246
$88$	0	0
$89$	15.7321	1.66759	0.833797	−	0.552071i	$-0.186162\pi$
$89$	15.7321	1.66759	0.833797	+	0.552071i	$0.186162\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−2.73205	−0.283300
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	8.12436	0.824903	0.412452	−	0.910979i	$-0.364673\pi$
$97$	8.12436	0.824903	0.412452	+	0.910979i	$0.364673\pi$
$98$	0	0
$99$	1.80385	0.181294
$100$	0	0
$101$	−0.196152	−0.0195179	−0.00975895	−	0.999952i	$-0.503106\pi$
$101$	−0.196152	−0.0195179	−0.00975895	+	0.999952i	$0.503106\pi$
$102$	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
$105$	1.26795	0.123739
$106$	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
$109$	−14.1962	−1.35974	−0.679872	−	0.733330i	$-0.737965\pi$
$109$	−14.1962	−1.35974	−0.679872	+	0.733330i	$0.737965\pi$
$110$	0	0
$111$	4.92820	0.467764
$112$	0	0
$113$	8.46410	0.796236	0.398118	−	0.917334i	$-0.369664\pi$
$113$	8.46410	0.796236	0.398118	+	0.917334i	$0.369664\pi$
$114$	0	0
$115$	−11.1962	−1.04405
$116$	0	0
$117$	2.46410	0.227806
$118$	0	0
$119$	−0.732051	−0.0671070
$120$	0	0
$121$	−10.4641	−0.951282
$122$	0	0
$123$	−7.60770	−0.685963
$124$	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
$129$	−4.05256	−0.356808
$130$	0	0
$131$	−13.4641	−1.17636	−0.588182	−	0.808729i	$-0.700156\pi$
$131$	−13.4641	−1.17636	−0.588182	+	0.808729i	$0.700156\pi$
$132$	0	0
$133$	1.73205	0.150188
$134$	0	0
$135$	−6.92820	−0.596285
$136$	0	0
$137$	10.7321	0.916901	0.458450	−	0.888720i	$-0.348405\pi$
$137$	10.7321	0.916901	0.458450	+	0.888720i	$0.348405\pi$
$138$	0	0
$139$	12.5885	1.06774	0.533870	−	0.845567i	$-0.320737\pi$
$139$	12.5885	1.06774	0.533870	+	0.845567i	$0.320737\pi$
$140$	0	0
$141$	−1.66025	−0.139819
$142$	0	0
$143$	0.732051	0.0612172
$144$	0	0
$145$	−0.803848	−0.0667559
$146$	0	0
$147$	−0.732051	−0.0603785
$148$	0	0
$149$	0.339746	0.0278331	0.0139165	−	0.999903i	$-0.495570\pi$
$149$	0.339746	0.0278331	0.0139165	+	0.999903i	$0.495570\pi$
$150$	0	0
$151$	−9.66025	−0.786140	−0.393070	−	0.919508i	$-0.628587\pi$
$151$	−9.66025	−0.786140	−0.393070	+	0.919508i	$0.628587\pi$
$152$	0	0
$153$	1.80385	0.145832
$154$	0	0
$155$	−6.46410	−0.519209
$156$	0	0
$157$	3.66025	0.292120	0.146060	−	0.989276i	$-0.453341\pi$
$157$	3.66025	0.292120	0.146060	+	0.989276i	$0.453341\pi$
$158$	0	0
$159$	−2.87564	−0.228053
$160$	0	0
$161$	6.46410	0.509443
$162$	0	0
$163$	2.53590	0.198627	0.0993134	−	0.995056i	$-0.468335\pi$
$163$	2.53590	0.198627	0.0993134	+	0.995056i	$0.468335\pi$
$164$	0	0
$165$	−0.928203	−0.0722605
$166$	0	0
$167$	−5.19615	−0.402090	−0.201045	−	0.979582i	$-0.564434\pi$
$167$	−5.19615	−0.402090	−0.201045	+	0.979582i	$0.564434\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.26795	−0.326378
$172$	0	0
$173$	−1.80385	−0.137144	−0.0685720	−	0.997646i	$-0.521844\pi$
$173$	−1.80385	−0.137144	−0.0685720	+	0.997646i	$0.521844\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−2.53590	−0.190610
$178$	0	0
$179$	7.39230	0.552527	0.276263	−	0.961082i	$-0.410904\pi$
$179$	7.39230	0.552527	0.276263	+	0.961082i	$0.410904\pi$
$180$	0	0
$181$	−15.1244	−1.12418	−0.562092	−	0.827075i	$-0.690003\pi$
$181$	−15.1244	−1.12418	−0.562092	+	0.827075i	$0.690003\pi$
$182$	0	0
$183$	1.46410	0.108230
$184$	0	0
$185$	11.6603	0.857279
$186$	0	0
$187$	0.535898	0.0391888
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	−19.8564	−1.43676	−0.718380	−	0.695651i	$-0.755116\pi$
$191$	−19.8564	−1.43676	−0.718380	+	0.695651i	$0.755116\pi$
$192$	0	0
$193$	−4.53590	−0.326501	−0.163251	−	0.986585i	$-0.552198\pi$
$193$	−4.53590	−0.326501	−0.163251	+	0.986585i	$0.552198\pi$
$194$	0	0
$195$	−1.26795	−0.0907997
$196$	0	0
$197$	−4.92820	−0.351120	−0.175560	−	0.984469i	$-0.556174\pi$
$197$	−4.92820	−0.351120	−0.175560	+	0.984469i	$0.556174\pi$
$198$	0	0
$199$	3.66025	0.259469	0.129734	−	0.991549i	$-0.458588\pi$
$199$	3.66025	0.259469	0.129734	+	0.991549i	$0.458588\pi$
$200$	0	0
$201$	7.60770	0.536605
$202$	0	0
$203$	0.464102	0.0325735
$204$	0	0
$205$	−18.0000	−1.25717
$206$	0	0
$207$	−15.9282	−1.10709
$208$	0	0
$209$	−1.26795	−0.0877059
$210$	0	0
$211$	−19.7846	−1.36203	−0.681014	−	0.732270i	$-0.738461\pi$
$211$	−19.7846	−1.36203	−0.681014	+	0.732270i	$0.738461\pi$
$212$	0	0
$213$	−0.143594	−0.00983887
$214$	0	0
$215$	−9.58846	−0.653927
$216$	0	0
$217$	3.73205	0.253348
$218$	0	0
$219$	−4.58846	−0.310059
$220$	0	0
$221$	0.732051	0.0492431
$222$	0	0
$223$	−2.66025	−0.178144	−0.0890719	−	0.996025i	$-0.528390\pi$
$223$	−2.66025	−0.178144	−0.0890719	+	0.996025i	$0.528390\pi$
$224$	0	0
$225$	4.92820	0.328547
$226$	0	0
$227$	−9.46410	−0.628154	−0.314077	−	0.949397i	$-0.601695\pi$
$227$	−9.46410	−0.628154	−0.314077	+	0.949397i	$0.601695\pi$
$228$	0	0
$229$	2.92820	0.193501	0.0967506	−	0.995309i	$-0.469155\pi$
$229$	2.92820	0.193501	0.0967506	+	0.995309i	$0.469155\pi$
$230$	0	0
$231$	0.535898	0.0352595
$232$	0	0
$233$	−17.7846	−1.16511	−0.582554	−	0.812792i	$-0.697947\pi$
$233$	−17.7846	−1.16511	−0.582554	+	0.812792i	$0.697947\pi$
$234$	0	0
$235$	−3.92820	−0.256248
$236$	0	0
$237$	−7.26795	−0.472104
$238$	0	0
$239$	14.5359	0.940249	0.470125	−	0.882600i	$-0.344209\pi$
$239$	14.5359	0.940249	0.470125	+	0.882600i	$0.344209\pi$
$240$	0	0
$241$	28.6603	1.84617	0.923085	−	0.384597i	$-0.125660\pi$
$241$	28.6603	1.84617	0.923085	+	0.384597i	$0.125660\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	−1.73205	−0.110657
$246$	0	0
$247$	−1.73205	−0.110208
$248$	0	0
$249$	−7.12436	−0.451488
$250$	0	0
$251$	24.5885	1.55201	0.776005	−	0.630727i	$-0.217243\pi$
$251$	24.5885	1.55201	0.776005	+	0.630727i	$0.217243\pi$
$252$	0	0
$253$	−4.73205	−0.297501
$254$	0	0
$255$	−0.928203	−0.0581263
$256$	0	0
$257$	10.7321	0.669447	0.334723	−	0.942316i	$-0.391357\pi$
$257$	10.7321	0.669447	0.334723	+	0.942316i	$0.391357\pi$
$258$	0	0
$259$	−6.73205	−0.418309
$260$	0	0
$261$	−1.14359	−0.0707867
$262$	0	0
$263$	8.32051	0.513065	0.256532	−	0.966536i	$-0.417420\pi$
$263$	8.32051	0.513065	0.256532	+	0.966536i	$0.417420\pi$
$264$	0	0
$265$	−6.80385	−0.417957
$266$	0	0
$267$	−11.5167	−0.704808
$268$	0	0
$269$	15.3205	0.934108	0.467054	−	0.884229i	$-0.345315\pi$
$269$	15.3205	0.934108	0.467054	+	0.884229i	$0.345315\pi$
$270$	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
$273$	0.732051	0.0443057
$274$	0	0
$275$	1.46410	0.0882886
$276$	0	0
$277$	−25.5359	−1.53430	−0.767152	−	0.641466i	$-0.778327\pi$
$277$	−25.5359	−1.53430	−0.767152	+	0.641466i	$0.778327\pi$
$278$	0	0
$279$	−9.19615	−0.550559
$280$	0	0
$281$	−6.58846	−0.393034	−0.196517	−	0.980500i	$-0.562963\pi$
$281$	−6.58846	−0.393034	−0.196517	+	0.980500i	$0.562963\pi$
$282$	0	0
$283$	21.4641	1.27591	0.637954	−	0.770074i	$-0.279781\pi$
$283$	21.4641	1.27591	0.637954	+	0.770074i	$0.279781\pi$
$284$	0	0
$285$	2.19615	0.130089
$286$	0	0
$287$	10.3923	0.613438
$288$	0	0
$289$	−16.4641	−0.968477
$290$	0	0
$291$	−5.94744	−0.348645
$292$	0	0
$293$	−12.5167	−0.731231	−0.365616	−	0.930766i	$-0.619142\pi$
$293$	−12.5167	−0.731231	−0.365616	+	0.930766i	$0.619142\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−2.92820	−0.169912
$298$	0	0
$299$	−6.46410	−0.373829
$300$	0	0
$301$	5.53590	0.319084
$302$	0	0
$303$	0.143594	0.00824923
$304$	0	0
$305$	3.46410	0.198354
$306$	0	0
$307$	23.5885	1.34626	0.673132	−	0.739522i	$-0.264948\pi$
$307$	23.5885	1.34626	0.673132	+	0.739522i	$0.264948\pi$
$308$	0	0
$309$	2.92820	0.166580
$310$	0	0
$311$	−4.19615	−0.237942	−0.118971	−	0.992898i	$-0.537960\pi$
$311$	−4.19615	−0.237942	−0.118971	+	0.992898i	$0.537960\pi$
$312$	0	0
$313$	−32.9808	−1.86418	−0.932091	−	0.362223i	$-0.882018\pi$
$313$	−32.9808	−1.86418	−0.932091	+	0.362223i	$0.882018\pi$
$314$	0	0
$315$	4.26795	0.240472
$316$	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
$319$	−0.339746	−0.0190221
$320$	0	0
$321$	−7.32051	−0.408591
$322$	0	0
$323$	−1.26795	−0.0705506
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	10.3923	0.574696
$328$	0	0
$329$	2.26795	0.125036
$330$	0	0
$331$	22.7846	1.25236	0.626178	−	0.779680i	$-0.284618\pi$
$331$	22.7846	1.25236	0.626178	+	0.779680i	$0.284618\pi$
$332$	0	0
$333$	16.5885	0.909042
$334$	0	0
$335$	18.0000	0.983445
$336$	0	0
$337$	−7.53590	−0.410507	−0.205253	−	0.978709i	$-0.565802\pi$
$337$	−7.53590	−0.410507	−0.205253	+	0.978709i	$0.565802\pi$
$338$	0	0
$339$	−6.19615	−0.336529
$340$	0	0
$341$	−2.73205	−0.147949
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	8.19615	0.441266
$346$	0	0
$347$	−21.0718	−1.13119	−0.565597	−	0.824682i	$-0.691354\pi$
$347$	−21.0718	−1.13119	−0.565597	+	0.824682i	$0.691354\pi$
$348$	0	0
$349$	17.5885	0.941489	0.470744	−	0.882270i	$-0.343985\pi$
$349$	17.5885	0.941489	0.470744	+	0.882270i	$0.343985\pi$
$350$	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
$355$	−0.339746	−0.0180318
$356$	0	0
$357$	0.535898	0.0283628
$358$	0	0
$359$	27.5167	1.45227	0.726137	−	0.687550i	$-0.241314\pi$
$359$	27.5167	1.45227	0.726137	+	0.687550i	$0.241314\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	0	0
$363$	7.66025	0.402059
$364$	0	0
$365$	−10.8564	−0.568250
$366$	0	0
$367$	4.87564	0.254507	0.127253	−	0.991870i	$-0.459384\pi$
$367$	4.87564	0.254507	0.127253	+	0.991870i	$0.459384\pi$
$368$	0	0
$369$	−25.6077	−1.33308
$370$	0	0
$371$	3.92820	0.203942
$372$	0	0
$373$	5.85641	0.303233	0.151617	−	0.988439i	$-0.451552\pi$
$373$	5.85641	0.303233	0.151617	+	0.988439i	$0.451552\pi$
$374$	0	0
$375$	−8.87564	−0.458336
$376$	0	0
$377$	−0.464102	−0.0239024
$378$	0	0
$379$	26.5885	1.36576	0.682879	−	0.730532i	$-0.260728\pi$
$379$	26.5885	1.36576	0.682879	+	0.730532i	$0.260728\pi$
$380$	0	0
$381$	−10.5359	−0.539770
$382$	0	0
$383$	−10.5359	−0.538359	−0.269180	−	0.963090i	$-0.586753\pi$
$383$	−10.5359	−0.538359	−0.269180	+	0.963090i	$0.586753\pi$
$384$	0	0
$385$	1.26795	0.0646207
$386$	0	0
$387$	−13.6410	−0.693412
$388$	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
$391$	−4.73205	−0.239310
$392$	0	0
$393$	9.85641	0.497190
$394$	0	0
$395$	−17.1962	−0.865232
$396$	0	0
$397$	−27.5885	−1.38462	−0.692312	−	0.721598i	$-0.743408\pi$
$397$	−27.5885	−1.38462	−0.692312	+	0.721598i	$0.743408\pi$
$398$	0	0
$399$	−1.26795	−0.0634769
$400$	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
$403$	−3.73205	−0.185907
$404$	0	0
$405$	−7.73205	−0.384209
$406$	0	0
$407$	4.92820	0.244282
$408$	0	0
$409$	20.5167	1.01448	0.507242	−	0.861804i	$-0.330665\pi$
$409$	20.5167	1.01448	0.507242	+	0.861804i	$0.330665\pi$
$410$	0	0
$411$	−7.85641	−0.387528
$412$	0	0
$413$	3.46410	0.170457
$414$	0	0
$415$	−16.8564	−0.827448
$416$	0	0
$417$	−9.21539	−0.451280
$418$	0	0
$419$	5.26795	0.257356	0.128678	−	0.991686i	$-0.458927\pi$
$419$	5.26795	0.257356	0.128678	+	0.991686i	$0.458927\pi$
$420$	0	0
$421$	−31.9090	−1.55515	−0.777574	−	0.628792i	$-0.783550\pi$
$421$	−31.9090	−1.55515	−0.777574	+	0.628792i	$0.783550\pi$
$422$	0	0
$423$	−5.58846	−0.271720
$424$	0	0
$425$	1.46410	0.0710194
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−0.535898	−0.0258734
$430$	0	0
$431$	−25.7128	−1.23854	−0.619271	−	0.785177i	$-0.712572\pi$
$431$	−25.7128	−1.23854	−0.619271	+	0.785177i	$0.712572\pi$
$432$	0	0
$433$	−1.07180	−0.0515073	−0.0257536	−	0.999668i	$-0.508199\pi$
$433$	−1.07180	−0.0515073	−0.0257536	+	0.999668i	$0.508199\pi$
$434$	0	0
$435$	0.588457	0.0282144
$436$	0	0
$437$	11.1962	0.535585
$438$	0	0
$439$	−8.24871	−0.393690	−0.196845	−	0.980435i	$-0.563069\pi$
$439$	−8.24871	−0.393690	−0.196845	+	0.980435i	$0.563069\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	29.7846	1.41511	0.707555	−	0.706659i	$-0.249798\pi$
$443$	29.7846	1.41511	0.707555	+	0.706659i	$0.249798\pi$
$444$	0	0
$445$	−27.2487	−1.29171
$446$	0	0
$447$	−0.248711	−0.0117636
$448$	0	0
$449$	24.3923	1.15114	0.575572	−	0.817751i	$-0.304779\pi$
$449$	24.3923	1.15114	0.575572	+	0.817751i	$0.304779\pi$
$450$	0	0
$451$	−7.60770	−0.358232
$452$	0	0
$453$	7.07180	0.332262
$454$	0	0
$455$	1.73205	0.0811998
$456$	0	0
$457$	2.73205	0.127800	0.0639000	−	0.997956i	$-0.479646\pi$
$457$	2.73205	0.127800	0.0639000	+	0.997956i	$0.479646\pi$
$458$	0	0
$459$	−2.92820	−0.136677
$460$	0	0
$461$	18.3923	0.856615	0.428308	−	0.903633i	$-0.359110\pi$
$461$	18.3923	0.856615	0.428308	+	0.903633i	$0.359110\pi$
$462$	0	0
$463$	22.5885	1.04977	0.524887	−	0.851172i	$-0.324107\pi$
$463$	22.5885	1.04977	0.524887	+	0.851172i	$0.324107\pi$
$464$	0	0
$465$	4.73205	0.219444
$466$	0	0
$467$	−24.5885	−1.13782	−0.568909	−	0.822400i	$-0.692634\pi$
$467$	−24.5885	−1.13782	−0.568909	+	0.822400i	$0.692634\pi$
$468$	0	0
$469$	−10.3923	−0.479872
$470$	0	0
$471$	−2.67949	−0.123464
$472$	0	0
$473$	−4.05256	−0.186337
$474$	0	0
$475$	−3.46410	−0.158944
$476$	0	0
$477$	−9.67949	−0.443193
$478$	0	0
$479$	40.1244	1.83333	0.916664	−	0.399658i	$-0.130871\pi$
$479$	40.1244	1.83333	0.916664	+	0.399658i	$0.130871\pi$
$480$	0	0
$481$	6.73205	0.306955
$482$	0	0
$483$	−4.73205	−0.215316
$484$	0	0
$485$	−14.0718	−0.638967
$486$	0	0
$487$	6.39230	0.289663	0.144831	−	0.989456i	$-0.453736\pi$
$487$	6.39230	0.289663	0.144831	+	0.989456i	$0.453736\pi$
$488$	0	0
$489$	−1.85641	−0.0839496
$490$	0	0
$491$	25.3205	1.14270	0.571349	−	0.820707i	$-0.306420\pi$
$491$	25.3205	1.14270	0.571349	+	0.820707i	$0.306420\pi$
$492$	0	0
$493$	−0.339746	−0.0153014
$494$	0	0
$495$	−3.12436	−0.140429
$496$	0	0
$497$	0.196152	0.00879864
$498$	0	0
$499$	15.5167	0.694621	0.347311	−	0.937750i	$-0.387095\pi$
$499$	15.5167	0.694621	0.347311	+	0.937750i	$0.387095\pi$
$500$	0	0
$501$	3.80385	0.169943
$502$	0	0
$503$	−36.6410	−1.63374	−0.816871	−	0.576820i	$-0.804293\pi$
$503$	−36.6410	−1.63374	−0.816871	+	0.576820i	$0.804293\pi$
$504$	0	0
$505$	0.339746	0.0151185
$506$	0	0
$507$	−0.732051	−0.0325115
$508$	0	0
$509$	18.5167	0.820737	0.410368	−	0.911920i	$-0.365400\pi$
$509$	18.5167	0.820737	0.410368	+	0.911920i	$0.365400\pi$
$510$	0	0
$511$	6.26795	0.277278
$512$	0	0
$513$	6.92820	0.305888
$514$	0	0
$515$	6.92820	0.305293
$516$	0	0
$517$	−1.66025	−0.0730179
$518$	0	0
$519$	1.32051	0.0579639
$520$	0	0
$521$	−16.7846	−0.735347	−0.367674	−	0.929955i	$-0.619846\pi$
$521$	−16.7846	−0.735347	−0.367674	+	0.929955i	$0.619846\pi$
$522$	0	0
$523$	−8.14359	−0.356094	−0.178047	−	0.984022i	$-0.556978\pi$
$523$	−8.14359	−0.356094	−0.178047	+	0.984022i	$0.556978\pi$
$524$	0	0
$525$	1.46410	0.0638986
$526$	0	0
$527$	−2.73205	−0.119010
$528$	0	0
$529$	18.7846	0.816722
$530$	0	0
$531$	−8.53590	−0.370426
$532$	0	0
$533$	−10.3923	−0.450141
$534$	0	0
$535$	−17.3205	−0.748831
$536$	0	0
$537$	−5.41154	−0.233525
$538$	0	0
$539$	−0.732051	−0.0315317
$540$	0	0
$541$	27.5167	1.18303	0.591517	−	0.806293i	$-0.298529\pi$
$541$	27.5167	1.18303	0.591517	+	0.806293i	$0.298529\pi$
$542$	0	0
$543$	11.0718	0.475136
$544$	0	0
$545$	24.5885	1.05325
$546$	0	0
$547$	−1.67949	−0.0718099	−0.0359049	−	0.999355i	$-0.511431\pi$
$547$	−1.67949	−0.0718099	−0.0359049	+	0.999355i	$0.511431\pi$
$548$	0	0
$549$	4.92820	0.210331
$550$	0	0
$551$	0.803848	0.0342451
$552$	0	0
$553$	9.92820	0.422190
$554$	0	0
$555$	−8.53590	−0.362329
$556$	0	0
$557$	−27.7128	−1.17423	−0.587115	−	0.809504i	$-0.699736\pi$
$557$	−27.7128	−1.17423	−0.587115	+	0.809504i	$0.699736\pi$
$558$	0	0
$559$	−5.53590	−0.234144
$560$	0	0
$561$	−0.392305	−0.0165631
$562$	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
$565$	−14.6603	−0.616762
$566$	0	0
$567$	4.46410	0.187475
$568$	0	0
$569$	24.4641	1.02559	0.512794	−	0.858512i	$-0.328610\pi$
$569$	24.4641	1.02559	0.512794	+	0.858512i	$0.328610\pi$
$570$	0	0
$571$	−16.1769	−0.676983	−0.338491	−	0.940969i	$-0.609917\pi$
$571$	−16.1769	−0.676983	−0.338491	+	0.940969i	$0.609917\pi$
$572$	0	0
$573$	14.5359	0.607246
$574$	0	0
$575$	−12.9282	−0.539143
$576$	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
$579$	3.32051	0.137996
$580$	0	0
$581$	9.73205	0.403754
$582$	0	0
$583$	−2.87564	−0.119097
$584$	0	0
$585$	−4.26795	−0.176458
$586$	0	0
$587$	21.3397	0.880786	0.440393	−	0.897805i	$-0.354839\pi$
$587$	21.3397	0.880786	0.440393	+	0.897805i	$0.354839\pi$
$588$	0	0
$589$	6.46410	0.266349
$590$	0	0
$591$	3.60770	0.148401
$592$	0	0
$593$	−1.05256	−0.0432234	−0.0216117	−	0.999766i	$-0.506880\pi$
$593$	−1.05256	−0.0432234	−0.0216117	+	0.999766i	$0.506880\pi$
$594$	0	0
$595$	1.26795	0.0519808
$596$	0	0
$597$	−2.67949	−0.109664
$598$	0	0
$599$	−25.7846	−1.05353	−0.526765	−	0.850011i	$-0.676595\pi$
$599$	−25.7846	−1.05353	−0.526765	+	0.850011i	$0.676595\pi$
$600$	0	0
$601$	23.8564	0.973123	0.486562	−	0.873646i	$-0.338251\pi$
$601$	23.8564	0.973123	0.486562	+	0.873646i	$0.338251\pi$
$602$	0	0
$603$	25.6077	1.04283
$604$	0	0
$605$	18.1244	0.736860
$606$	0	0
$607$	27.8038	1.12852	0.564262	−	0.825596i	$-0.309161\pi$
$607$	27.8038	1.12852	0.564262	+	0.825596i	$0.309161\pi$
$608$	0	0
$609$	−0.339746	−0.0137672
$610$	0	0
$611$	−2.26795	−0.0917514
$612$	0	0
$613$	6.14359	0.248137	0.124069	−	0.992274i	$-0.460406\pi$
$613$	6.14359	0.248137	0.124069	+	0.992274i	$0.460406\pi$
$614$	0	0
$615$	13.1769	0.531344
$616$	0	0
$617$	28.6410	1.15304	0.576522	−	0.817082i	$-0.304410\pi$
$617$	28.6410	1.15304	0.576522	+	0.817082i	$0.304410\pi$
$618$	0	0
$619$	19.7128	0.792325	0.396162	−	0.918180i	$-0.370342\pi$
$619$	19.7128	0.792325	0.396162	+	0.918180i	$0.370342\pi$
$620$	0	0
$621$	25.8564	1.03758
$622$	0	0
$623$	15.7321	0.630291
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0.928203	0.0370689
$628$	0	0
$629$	4.92820	0.196500
$630$	0	0
$631$	−25.7128	−1.02361	−0.511805	−	0.859101i	$-0.671023\pi$
$631$	−25.7128	−1.02361	−0.511805	+	0.859101i	$0.671023\pi$
$632$	0	0
$633$	14.4833	0.575661
$634$	0	0
$635$	−24.9282	−0.989246
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−0.483340	−0.0191206
$640$	0	0
$641$	43.6410	1.72372	0.861858	−	0.507149i	$-0.169301\pi$
$641$	43.6410	1.72372	0.861858	+	0.507149i	$0.169301\pi$
$642$	0	0
$643$	−17.1769	−0.677391	−0.338696	−	0.940896i	$-0.609986\pi$
$643$	−17.1769	−0.677391	−0.338696	+	0.940896i	$0.609986\pi$
$644$	0	0
$645$	7.01924	0.276382
$646$	0	0
$647$	24.5885	0.966672	0.483336	−	0.875435i	$-0.339425\pi$
$647$	24.5885	0.966672	0.483336	+	0.875435i	$0.339425\pi$
$648$	0	0
$649$	−2.53590	−0.0995427
$650$	0	0
$651$	−2.73205	−0.107078
$652$	0	0
$653$	−7.85641	−0.307445	−0.153722	−	0.988114i	$-0.549126\pi$
$653$	−7.85641	−0.307445	−0.153722	+	0.988114i	$0.549126\pi$
$654$	0	0
$655$	23.3205	0.911208
$656$	0	0
$657$	−15.4449	−0.602562
$658$	0	0
$659$	1.92820	0.0751121	0.0375561	−	0.999295i	$-0.488043\pi$
$659$	1.92820	0.0751121	0.0375561	+	0.999295i	$0.488043\pi$
$660$	0	0
$661$	31.1962	1.21339	0.606695	−	0.794935i	$-0.292495\pi$
$661$	31.1962	1.21339	0.606695	+	0.794935i	$0.292495\pi$
$662$	0	0
$663$	−0.535898	−0.0208126
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	3.00000	0.116160
$668$	0	0
$669$	1.94744	0.0752924
$670$	0	0
$671$	1.46410	0.0565210
$672$	0	0
$673$	27.3923	1.05590	0.527948	−	0.849277i	$-0.322962\pi$
$673$	27.3923	1.05590	0.527948	+	0.849277i	$0.322962\pi$
$674$	0	0
$675$	−8.00000	−0.307920
$676$	0	0
$677$	−20.9808	−0.806356	−0.403178	−	0.915122i	$-0.632094\pi$
$677$	−20.9808	−0.806356	−0.403178	+	0.915122i	$0.632094\pi$
$678$	0	0
$679$	8.12436	0.311784
$680$	0	0
$681$	6.92820	0.265489
$682$	0	0
$683$	1.46410	0.0560223	0.0280111	−	0.999608i	$-0.491083\pi$
$683$	1.46410	0.0560223	0.0280111	+	0.999608i	$0.491083\pi$
$684$	0	0
$685$	−18.5885	−0.710228
$686$	0	0
$687$	−2.14359	−0.0817832
$688$	0	0
$689$	−3.92820	−0.149653
$690$	0	0
$691$	37.3013	1.41901	0.709504	−	0.704702i	$-0.248919\pi$
$691$	37.3013	1.41901	0.709504	+	0.704702i	$0.248919\pi$
$692$	0	0
$693$	1.80385	0.0685225
$694$	0	0
$695$	−21.8038	−0.827067
$696$	0	0
$697$	−7.60770	−0.288162
$698$	0	0
$699$	13.0192	0.492433
$700$	0	0
$701$	28.3205	1.06965	0.534825	−	0.844963i	$-0.320377\pi$
$701$	28.3205	1.06965	0.534825	+	0.844963i	$0.320377\pi$
$702$	0	0
$703$	−11.6603	−0.439775
$704$	0	0
$705$	2.87564	0.108303
$706$	0	0
$707$	−0.196152	−0.00737707
$708$	0	0
$709$	7.94744	0.298472	0.149236	−	0.988802i	$-0.452319\pi$
$709$	7.94744	0.298472	0.149236	+	0.988802i	$0.452319\pi$
$710$	0	0
$711$	−24.4641	−0.917475
$712$	0	0
$713$	24.1244	0.903464
$714$	0	0
$715$	−1.26795	−0.0474186
$716$	0	0
$717$	−10.6410	−0.397396
$718$	0	0
$719$	22.7321	0.847762	0.423881	−	0.905718i	$-0.360667\pi$
$719$	22.7321	0.847762	0.423881	+	0.905718i	$0.360667\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	−20.9808	−0.780283
$724$	0	0
$725$	−0.928203	−0.0344726
$726$	0	0
$727$	−34.7846	−1.29009	−0.645045	−	0.764145i	$-0.723161\pi$
$727$	−34.7846	−1.29009	−0.645045	+	0.764145i	$0.723161\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−4.05256	−0.149889
$732$	0	0
$733$	−41.3013	−1.52550	−0.762749	−	0.646695i	$-0.776151\pi$
$733$	−41.3013	−1.52550	−0.762749	+	0.646695i	$0.776151\pi$
$734$	0	0
$735$	1.26795	0.0467690
$736$	0	0
$737$	7.60770	0.280233
$738$	0	0
$739$	−8.73205	−0.321214	−0.160607	−	0.987018i	$-0.551345\pi$
$739$	−8.73205	−0.321214	−0.160607	+	0.987018i	$0.551345\pi$
$740$	0	0
$741$	1.26795	0.0465793
$742$	0	0
$743$	10.1436	0.372132	0.186066	−	0.982537i	$-0.440426\pi$
$743$	10.1436	0.372132	0.186066	+	0.982537i	$0.440426\pi$
$744$	0	0
$745$	−0.588457	−0.0215594
$746$	0	0
$747$	−23.9808	−0.877410
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	9.92820	0.362285	0.181143	−	0.983457i	$-0.442020\pi$
$751$	9.92820	0.362285	0.181143	+	0.983457i	$0.442020\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	16.7321	0.608942
$756$	0	0
$757$	34.4641	1.25262	0.626310	−	0.779574i	$-0.284565\pi$
$757$	34.4641	1.25262	0.626310	+	0.779574i	$0.284565\pi$
$758$	0	0
$759$	3.46410	0.125739
$760$	0	0
$761$	39.7321	1.44029	0.720143	−	0.693826i	$-0.244076\pi$
$761$	39.7321	1.44029	0.720143	+	0.693826i	$0.244076\pi$
$762$	0	0
$763$	−14.1962	−0.513935
$764$	0	0
$765$	−3.12436	−0.112961
$766$	0	0
$767$	−3.46410	−0.125081
$768$	0	0
$769$	36.1244	1.30268	0.651339	−	0.758787i	$-0.274208\pi$
$769$	36.1244	1.30268	0.651339	+	0.758787i	$0.274208\pi$
$770$	0	0
$771$	−7.85641	−0.282942
$772$	0	0
$773$	19.4641	0.700075	0.350038	−	0.936736i	$-0.386169\pi$
$773$	19.4641	0.700075	0.350038	+	0.936736i	$0.386169\pi$
$774$	0	0
$775$	−7.46410	−0.268118
$776$	0	0
$777$	4.92820	0.176798
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	−0.143594	−0.00513818
$782$	0	0
$783$	1.85641	0.0663426
$784$	0	0
$785$	−6.33975	−0.226275
$786$	0	0
$787$	12.5167	0.446171	0.223085	−	0.974799i	$-0.428387\pi$
$787$	12.5167	0.446171	0.223085	+	0.974799i	$0.428387\pi$
$788$	0	0
$789$	−6.09103	−0.216847
$790$	0	0
$791$	8.46410	0.300949
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	4.98076	0.176649
$796$	0	0
$797$	−14.7846	−0.523698	−0.261849	−	0.965109i	$-0.584332\pi$
$797$	−14.7846	−0.523698	−0.261849	+	0.965109i	$0.584332\pi$
$798$	0	0
$799$	−1.66025	−0.0587356
$800$	0	0
$801$	−38.7654	−1.36971
$802$	0	0
$803$	−4.58846	−0.161923
$804$	0	0
$805$	−11.1962	−0.394613
$806$	0	0
$807$	−11.2154	−0.394800
$808$	0	0
$809$	−50.0333	−1.75908	−0.879539	−	0.475827i	$-0.842149\pi$
$809$	−50.0333	−1.75908	−0.879539	+	0.475827i	$0.842149\pi$
$810$	0	0
$811$	−28.7846	−1.01076	−0.505382	−	0.862896i	$-0.668648\pi$
$811$	−28.7846	−1.01076	−0.505382	+	0.862896i	$0.668648\pi$
$812$	0	0
$813$	8.78461	0.308090
$814$	0	0
$815$	−4.39230	−0.153856
$816$	0	0
$817$	9.58846	0.335458
$818$	0	0
$819$	2.46410	0.0861027
$820$	0	0
$821$	49.3205	1.72130	0.860649	−	0.509199i	$-0.170058\pi$
$821$	49.3205	1.72130	0.860649	+	0.509199i	$0.170058\pi$
$822$	0	0
$823$	20.1436	0.702162	0.351081	−	0.936345i	$-0.385814\pi$
$823$	20.1436	0.702162	0.351081	+	0.936345i	$0.385814\pi$
$824$	0	0
$825$	−1.07180	−0.0373152
$826$	0	0
$827$	−25.1769	−0.875487	−0.437744	−	0.899100i	$-0.644222\pi$
$827$	−25.1769	−0.875487	−0.437744	+	0.899100i	$0.644222\pi$
$828$	0	0
$829$	−5.94744	−0.206563	−0.103282	−	0.994652i	$-0.532934\pi$
$829$	−5.94744	−0.206563	−0.103282	+	0.994652i	$0.532934\pi$
$830$	0	0
$831$	18.6936	0.648473
$832$	0	0
$833$	−0.732051	−0.0253641
$834$	0	0
$835$	9.00000	0.311458
$836$	0	0
$837$	14.9282	0.515994
$838$	0	0
$839$	34.5359	1.19231	0.596156	−	0.802869i	$-0.296694\pi$
$839$	34.5359	1.19231	0.596156	+	0.802869i	$0.296694\pi$
$840$	0	0
$841$	−28.7846	−0.992573
$842$	0	0
$843$	4.82309	0.166116
$844$	0	0
$845$	−1.73205	−0.0595844
$846$	0	0
$847$	−10.4641	−0.359551
$848$	0	0
$849$	−15.7128	−0.539262
$850$	0	0
$851$	−43.5167	−1.49173
$852$	0	0
$853$	25.8756	0.885965	0.442983	−	0.896530i	$-0.353920\pi$
$853$	25.8756	0.885965	0.442983	+	0.896530i	$0.353920\pi$
$854$	0	0
$855$	7.39230	0.252811
$856$	0	0
$857$	−28.5359	−0.974768	−0.487384	−	0.873188i	$-0.662049\pi$
$857$	−28.5359	−0.974768	−0.487384	+	0.873188i	$0.662049\pi$
$858$	0	0
$859$	−35.5692	−1.21361	−0.606803	−	0.794852i	$-0.707548\pi$
$859$	−35.5692	−1.21361	−0.606803	+	0.794852i	$0.707548\pi$
$860$	0	0
$861$	−7.60770	−0.259270
$862$	0	0
$863$	−46.9282	−1.59745	−0.798727	−	0.601693i	$-0.794493\pi$
$863$	−46.9282	−1.59745	−0.798727	+	0.601693i	$0.794493\pi$
$864$	0	0
$865$	3.12436	0.106231
$866$	0	0
$867$	12.0526	0.409326
$868$	0	0
$869$	−7.26795	−0.246548
$870$	0	0
$871$	10.3923	0.352130
$872$	0	0
$873$	−20.0192	−0.677549
$874$	0	0
$875$	12.1244	0.409878
$876$	0	0
$877$	35.5167	1.19931	0.599656	−	0.800258i	$-0.295304\pi$
$877$	35.5167	1.19931	0.599656	+	0.800258i	$0.295304\pi$
$878$	0	0
$879$	9.16283	0.309055
$880$	0	0
$881$	14.6795	0.494565	0.247282	−	0.968943i	$-0.420462\pi$
$881$	14.6795	0.494565	0.247282	+	0.968943i	$0.420462\pi$
$882$	0	0
$883$	4.24871	0.142981	0.0714903	−	0.997441i	$-0.477225\pi$
$883$	4.24871	0.142981	0.0714903	+	0.997441i	$0.477225\pi$
$884$	0	0
$885$	4.39230	0.147646
$886$	0	0
$887$	39.0333	1.31061	0.655305	−	0.755364i	$-0.272540\pi$
$887$	39.0333	1.31061	0.655305	+	0.755364i	$0.272540\pi$
$888$	0	0
$889$	14.3923	0.482702
$890$	0	0
$891$	−3.26795	−0.109480
$892$	0	0
$893$	3.92820	0.131452
$894$	0	0
$895$	−12.8038	−0.427985
$896$	0	0
$897$	4.73205	0.157999
$898$	0	0
$899$	1.73205	0.0577671
$900$	0	0
$901$	−2.87564	−0.0958016
$902$	0	0
$903$	−4.05256	−0.134861
$904$	0	0
$905$	26.1962	0.870790
$906$	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
$909$	0.483340	0.0160314
$910$	0	0
$911$	−5.24871	−0.173898	−0.0869488	−	0.996213i	$-0.527712\pi$
$911$	−5.24871	−0.173898	−0.0869488	+	0.996213i	$0.527712\pi$
$912$	0	0
$913$	−7.12436	−0.235782
$914$	0	0
$915$	−2.53590	−0.0838342
$916$	0	0
$917$	−13.4641	−0.444624
$918$	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
$921$	−17.2679	−0.568998
$922$	0	0
$923$	−0.196152	−0.00645644
$924$	0	0
$925$	13.4641	0.442697
$926$	0	0
$927$	9.85641	0.323727
$928$	0	0
$929$	−25.4449	−0.834819	−0.417409	−	0.908719i	$-0.637062\pi$
$929$	−25.4449	−0.834819	−0.417409	+	0.908719i	$0.637062\pi$
$930$	0	0
$931$	1.73205	0.0567657
$932$	0	0
$933$	3.07180	0.100566
$934$	0	0
$935$	−0.928203	−0.0303555
$936$	0	0
$937$	−52.8372	−1.72612	−0.863058	−	0.505106i	$-0.831454\pi$
$937$	−52.8372	−1.72612	−0.863058	+	0.505106i	$0.831454\pi$
$938$	0	0
$939$	24.1436	0.787896
$940$	0	0
$941$	−29.1962	−0.951767	−0.475884	−	0.879508i	$-0.657872\pi$
$941$	−29.1962	−0.951767	−0.475884	+	0.879508i	$0.657872\pi$
$942$	0	0
$943$	67.1769	2.18758
$944$	0	0
$945$	−6.92820	−0.225374
$946$	0	0
$947$	41.3205	1.34274	0.671368	−	0.741124i	$-0.265707\pi$
$947$	41.3205	1.34274	0.671368	+	0.741124i	$0.265707\pi$
$948$	0	0
$949$	−6.26795	−0.203466
$950$	0	0
$951$	20.4974	0.664674
$952$	0	0
$953$	−17.5359	−0.568043	−0.284022	−	0.958818i	$-0.591669\pi$
$953$	−17.5359	−0.568043	−0.284022	+	0.958818i	$0.591669\pi$
$954$	0	0
$955$	34.3923	1.11291
$956$	0	0
$957$	0.248711	0.00803969
$958$	0	0
$959$	10.7321	0.346556
$960$	0	0
$961$	−17.0718	−0.550703
$962$	0	0
$963$	−24.6410	−0.794046
$964$	0	0
$965$	7.85641	0.252907
$966$	0	0
$967$	−42.9808	−1.38217	−0.691084	−	0.722774i	$-0.742867\pi$
$967$	−42.9808	−1.38217	−0.691084	+	0.722774i	$0.742867\pi$
$968$	0	0
$969$	0.928203	0.0298182
$970$	0	0
$971$	46.3923	1.48880	0.744400	−	0.667734i	$-0.232736\pi$
$971$	46.3923	1.48880	0.744400	+	0.667734i	$0.232736\pi$
$972$	0	0
$973$	12.5885	0.403567
$974$	0	0
$975$	−1.46410	−0.0468888
$976$	0	0
$977$	10.5885	0.338755	0.169377	−	0.985551i	$-0.445824\pi$
$977$	10.5885	0.338755	0.169377	+	0.985551i	$0.445824\pi$
$978$	0	0
$979$	−11.5167	−0.368074
$980$	0	0
$981$	34.9808	1.11685
$982$	0	0
$983$	24.2679	0.774027	0.387014	−	0.922074i	$-0.373507\pi$
$983$	24.2679	0.774027	0.387014	+	0.922074i	$0.373507\pi$
$984$	0	0
$985$	8.53590	0.271976
$986$	0	0
$987$	−1.66025	−0.0528465
$988$	0	0
$989$	35.7846	1.13788
$990$	0	0
$991$	−2.39230	−0.0759941	−0.0379970	−	0.999278i	$-0.512098\pi$
$991$	−2.39230	−0.0759941	−0.0379970	+	0.999278i	$0.512098\pi$
$992$	0	0
$993$	−16.6795	−0.529308
$994$	0	0
$995$	−6.33975	−0.200983
$996$	0	0
$997$	12.3923	0.392468	0.196234	−	0.980557i	$-0.437129\pi$
$997$	12.3923	0.392468	0.196234	+	0.980557i	$0.437129\pi$
$998$	0	0
$999$	−26.9282	−0.851971

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} -1.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} -1.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} -0.732051 q^{11} -1.00000 q^{13} +1.26795 q^{15} -0.732051 q^{17} +1.73205 q^{19} -0.732051 q^{21} +6.46410 q^{23} -2.00000 q^{25} +4.00000 q^{27} +0.464102 q^{29} +3.73205 q^{31} +0.535898 q^{33} -1.73205 q^{35} -6.73205 q^{37} +0.732051 q^{39} +10.3923 q^{41} +5.53590 q^{43} +4.26795 q^{45} +2.26795 q^{47} +1.00000 q^{49} +0.535898 q^{51} +3.92820 q^{53} +1.26795 q^{55} -1.26795 q^{57} +3.46410 q^{59} -2.00000 q^{61} -2.46410 q^{63} +1.73205 q^{65} -10.3923 q^{67} -4.73205 q^{69} +0.196152 q^{71} +6.26795 q^{73} +1.46410 q^{75} -0.732051 q^{77} +9.92820 q^{79} +4.46410 q^{81} +9.73205 q^{83} +1.26795 q^{85} -0.339746 q^{87} +15.7321 q^{89} -1.00000 q^{91} -2.73205 q^{93} -3.00000 q^{95} +8.12436 q^{97} +1.80385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more