LMFDB - Embedded newform 1856.2.a.ba.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.230224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.49889$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86312 q^{3} +0.199646 q^{5} +4.99779 q^{7} +0.471201 q^{9} +O(q^{10})$	$q+1.86312 q^{3} +0.199646 q^{5} +4.99779 q^{7} +0.471201 q^{9} -3.95719 q^{11} +1.70628 q^{13} +0.371963 q^{15} -0.329153 q^{17} +6.48894 q^{19} +9.31146 q^{21} +2.72844 q^{23} -4.96014 q^{25} -4.71145 q^{27} +1.00000 q^{29} +10.1170 q^{31} -7.37270 q^{33} +0.997788 q^{35} -4.58301 q^{37} +3.17900 q^{39} +10.7240 q^{41} -3.95719 q^{43} +0.0940734 q^{45} -0.954977 q^{47} +17.9779 q^{49} -0.613250 q^{51} +2.29372 q^{53} -0.790036 q^{55} +12.0896 q^{57} -12.7240 q^{59} +2.87226 q^{61} +2.35496 q^{63} +0.340652 q^{65} +6.98231 q^{67} +5.08341 q^{69} -6.92765 q^{71} -3.92471 q^{73} -9.24132 q^{75} -19.7772 q^{77} -6.49749 q^{79} -10.1916 q^{81} -12.2539 q^{83} -0.0657140 q^{85} +1.86312 q^{87} +14.9122 q^{89} +8.52763 q^{91} +18.8491 q^{93} +1.29549 q^{95} +3.74171 q^{97} -1.86463 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9	$ 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9} - 8 q^{11} + 6 q^{13} + 6 q^{15} + 6 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{23} + 7 q^{25} - 22 q^{27} + 5 q^{29} + 8 q^{31} - 16 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} + 28 q^{47} + 13 q^{49} - 24 q^{51} + 14 q^{53} + 10 q^{55} + 20 q^{57} - 16 q^{59} + 18 q^{61} + 20 q^{63} - 8 q^{65} + 28 q^{69} - 4 q^{71} + 2 q^{73} + 6 q^{75} + 4 q^{77} - 12 q^{79} + 9 q^{81} - 32 q^{83} + 32 q^{85} - 4 q^{87} + 30 q^{89} + 40 q^{91} - 4 q^{93} + 4 q^{95} + 6 q^{97} + 22 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9 - 8 * q^11 + 6 * q^13 + 6 * q^15 + 6 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^23 + 7 * q^25 - 22 * q^27 + 5 * q^29 + 8 * q^31 - 16 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 + 28 * q^47 + 13 * q^49 - 24 * q^51 + 14 * q^53 + 10 * q^55 + 20 * q^57 - 16 * q^59 + 18 * q^61 + 20 * q^63 - 8 * q^65 + 28 * q^69 - 4 * q^71 + 2 * q^73 + 6 * q^75 + 4 * q^77 - 12 * q^79 + 9 * q^81 - 32 * q^83 + 32 * q^85 - 4 * q^87 + 30 * q^89 + 40 * q^91 - 4 * q^93 + 4 * q^95 + 6 * q^97 + 22 * 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
$3$	1.86312	1.07567	0.537835	−	0.843050i	$-0.319242\pi$
$3$	1.86312	1.07567	0.537835	+	0.843050i	$0.319242\pi$
$4$	0	0
$5$	0.199646	0.0892843	0.0446422	−	0.999003i	$-0.485785\pi$
$5$	0.199646	0.0892843	0.0446422	+	0.999003i	$0.485785\pi$
$6$	0	0
$7$	4.99779	1.88899	0.944493	−	0.328531i	$-0.106554\pi$
$7$	4.99779	1.88899	0.944493	+	0.328531i	$0.106554\pi$
$8$	0	0
$9$	0.471201	0.157067
$10$	0	0
$11$	−3.95719	−1.19314	−0.596569	−	0.802562i	$-0.703470\pi$
$11$	−3.95719	−1.19314	−0.596569	+	0.802562i	$0.703470\pi$
$12$	0	0
$13$	1.70628	0.473237	0.236619	−	0.971603i	$-0.423961\pi$
$13$	1.70628	0.473237	0.236619	+	0.971603i	$0.423961\pi$
$14$	0	0
$15$	0.371963	0.0960405
$16$	0	0
$17$	−0.329153	−0.0798313	−0.0399156	−	0.999203i	$-0.512709\pi$
$17$	−0.329153	−0.0798313	−0.0399156	+	0.999203i	$0.512709\pi$
$18$	0	0
$19$	6.48894	1.48866	0.744332	−	0.667809i	$-0.232768\pi$
$19$	6.48894	1.48866	0.744332	+	0.667809i	$0.232768\pi$
$20$	0	0
$21$	9.31146	2.03193
$22$	0	0
$23$	2.72844	0.568920	0.284460	−	0.958688i	$-0.408186\pi$
$23$	2.72844	0.568920	0.284460	+	0.958688i	$0.408186\pi$
$24$	0	0
$25$	−4.96014	−0.992028
$26$	0	0
$27$	−4.71145	−0.906718
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	10.1170	1.81706	0.908531	−	0.417817i	$-0.137205\pi$
$31$	10.1170	1.81706	0.908531	+	0.417817i	$0.137205\pi$
$32$	0	0
$33$	−7.37270	−1.28342
$34$	0	0
$35$	0.997788	0.168657
$36$	0	0
$37$	−4.58301	−0.753443	−0.376721	−	0.926327i	$-0.622949\pi$
$37$	−4.58301	−0.753443	−0.376721	+	0.926327i	$0.622949\pi$
$38$	0	0
$39$	3.17900	0.509047
$40$	0	0
$41$	10.7240	1.67481	0.837405	−	0.546583i	$-0.184072\pi$
$41$	10.7240	1.67481	0.837405	+	0.546583i	$0.184072\pi$
$42$	0	0
$43$	−3.95719	−0.603466	−0.301733	−	0.953393i	$-0.597565\pi$
$43$	−3.95719	−0.603466	−0.301733	+	0.953393i	$0.597565\pi$
$44$	0	0
$45$	0.0940734	0.0140236
$46$	0	0
$47$	−0.954977	−0.139298	−0.0696489	−	0.997572i	$-0.522188\pi$
$47$	−0.954977	−0.139298	−0.0696489	+	0.997572i	$0.522188\pi$
$48$	0	0
$49$	17.9779	2.56827
$50$	0	0
$51$	−0.613250	−0.0858722
$52$	0	0
$53$	2.29372	0.315067	0.157533	−	0.987514i	$-0.449646\pi$
$53$	2.29372	0.315067	0.157533	+	0.987514i	$0.449646\pi$
$54$	0	0
$55$	−0.790036	−0.106528
$56$	0	0
$57$	12.0896	1.60131
$58$	0	0
$59$	−12.7240	−1.65653	−0.828263	−	0.560339i	$-0.810671\pi$
$59$	−12.7240	−1.65653	−0.828263	+	0.560339i	$0.810671\pi$
$60$	0	0
$61$	2.87226	0.367756	0.183878	−	0.982949i	$-0.441135\pi$
$61$	2.87226	0.367756	0.183878	+	0.982949i	$0.441135\pi$
$62$	0	0
$63$	2.35496	0.296698
$64$	0	0
$65$	0.340652	0.0422527
$66$	0	0
$67$	6.98231	0.853024	0.426512	−	0.904482i	$-0.359742\pi$
$67$	6.98231	0.853024	0.426512	+	0.904482i	$0.359742\pi$
$68$	0	0
$69$	5.08341	0.611970
$70$	0	0
$71$	−6.92765	−0.822161	−0.411080	−	0.911599i	$-0.634848\pi$
$71$	−6.92765	−0.822161	−0.411080	+	0.911599i	$0.634848\pi$
$72$	0	0
$73$	−3.92471	−0.459352	−0.229676	−	0.973267i	$-0.573767\pi$
$73$	−3.92471	−0.459352	−0.229676	+	0.973267i	$0.573767\pi$
$74$	0	0
$75$	−9.24132	−1.06710
$76$	0	0
$77$	−19.7772	−2.25382
$78$	0	0
$79$	−6.49749	−0.731024	−0.365512	−	0.930807i	$-0.619106\pi$
$79$	−6.49749	−0.731024	−0.365512	+	0.930807i	$0.619106\pi$
$80$	0	0
$81$	−10.1916	−1.13240
$82$	0	0
$83$	−12.2539	−1.34504	−0.672518	−	0.740081i	$-0.734787\pi$
$83$	−12.2539	−1.34504	−0.672518	+	0.740081i	$0.734787\pi$
$84$	0	0
$85$	−0.0657140	−0.00712768
$86$	0	0
$87$	1.86312	0.199747
$88$	0	0
$89$	14.9122	1.58069	0.790343	−	0.612664i	$-0.209902\pi$
$89$	14.9122	1.58069	0.790343	+	0.612664i	$0.209902\pi$
$90$	0	0
$91$	8.52763	0.893938
$92$	0	0
$93$	18.8491	1.95456
$94$	0	0
$95$	1.29549	0.132914
$96$	0	0
$97$	3.74171	0.379914	0.189957	−	0.981792i	$-0.439165\pi$
$97$	3.74171	0.379914	0.189957	+	0.981792i	$0.439165\pi$
$98$	0	0
$99$	−1.86463	−0.187403
$100$	0	0
$101$	−10.3904	−1.03389	−0.516944	−	0.856019i	$-0.672930\pi$
$101$	−10.3904	−1.03389	−0.516944	+	0.856019i	$0.672930\pi$
$102$	0	0
$103$	−7.06793	−0.696423	−0.348212	−	0.937416i	$-0.613211\pi$
$103$	−7.06793	−0.696423	−0.348212	+	0.937416i	$0.613211\pi$
$104$	0	0
$105$	1.85899	0.181419
$106$	0	0
$107$	−8.80964	−0.851660	−0.425830	−	0.904803i	$-0.640018\pi$
$107$	−8.80964	−0.851660	−0.425830	+	0.904803i	$0.640018\pi$
$108$	0	0
$109$	13.3657	1.28020	0.640100	−	0.768292i	$-0.278893\pi$
$109$	13.3657	1.28020	0.640100	+	0.768292i	$0.278893\pi$
$110$	0	0
$111$	−8.53869	−0.810456
$112$	0	0
$113$	−1.01327	−0.0953204	−0.0476602	−	0.998864i	$-0.515176\pi$
$113$	−1.01327	−0.0953204	−0.0476602	+	0.998864i	$0.515176\pi$
$114$	0	0
$115$	0.544723	0.0507956
$116$	0	0
$117$	0.804002	0.0743300
$118$	0	0
$119$	−1.64504	−0.150800
$120$	0	0
$121$	4.65935	0.423577
$122$	0	0
$123$	19.9801	1.80154
$124$	0	0
$125$	−1.98850	−0.177857
$126$	0	0
$127$	3.69478	0.327859	0.163929	−	0.986472i	$-0.447583\pi$
$127$	3.69478	0.327859	0.163929	+	0.986472i	$0.447583\pi$
$128$	0	0
$129$	−7.37270	−0.649130
$130$	0	0
$131$	−4.16494	−0.363893	−0.181946	−	0.983308i	$-0.558240\pi$
$131$	−4.16494	−0.363893	−0.181946	+	0.983308i	$0.558240\pi$
$132$	0	0
$133$	32.4303	2.81207
$134$	0	0
$135$	−0.940621	−0.0809557
$136$	0	0
$137$	−11.5963	−0.990737	−0.495369	−	0.868683i	$-0.664967\pi$
$137$	−11.5963	−0.990737	−0.495369	+	0.868683i	$0.664967\pi$
$138$	0	0
$139$	4.47297	0.379393	0.189696	−	0.981843i	$-0.439250\pi$
$139$	4.47297	0.379393	0.189696	+	0.981843i	$0.439250\pi$
$140$	0	0
$141$	−1.77923	−0.149838
$142$	0	0
$143$	−6.75208	−0.564637
$144$	0	0
$145$	0.199646	0.0165797
$146$	0	0
$147$	33.4949	2.76261
$148$	0	0
$149$	20.7994	1.70395	0.851975	−	0.523583i	$-0.175405\pi$
$149$	20.7994	1.70395	0.851975	+	0.523583i	$0.175405\pi$
$150$	0	0
$151$	16.3306	1.32897	0.664484	−	0.747302i	$-0.268651\pi$
$151$	16.3306	1.32897	0.664484	+	0.747302i	$0.268651\pi$
$152$	0	0
$153$	−0.155097	−0.0125389
$154$	0	0
$155$	2.01981	0.162235
$156$	0	0
$157$	−18.1765	−1.45064	−0.725321	−	0.688411i	$-0.758308\pi$
$157$	−18.1765	−1.45064	−0.725321	+	0.688411i	$0.758308\pi$
$158$	0	0
$159$	4.27347	0.338908
$160$	0	0
$161$	13.6362	1.07468
$162$	0	0
$163$	−14.1827	−1.11087	−0.555437	−	0.831559i	$-0.687449\pi$
$163$	−14.1827	−1.11087	−0.555437	+	0.831559i	$0.687449\pi$
$164$	0	0
$165$	−1.47193	−0.114590
$166$	0	0
$167$	15.7262	1.21693	0.608466	−	0.793580i	$-0.291785\pi$
$167$	15.7262	1.21693	0.608466	+	0.793580i	$0.291785\pi$
$168$	0	0
$169$	−10.0886	−0.776047
$170$	0	0
$171$	3.05760	0.233820
$172$	0	0
$173$	−4.54311	−0.345406	−0.172703	−	0.984974i	$-0.555250\pi$
$173$	−4.54311	−0.345406	−0.172703	+	0.984974i	$0.555250\pi$
$174$	0	0
$175$	−24.7897	−1.87393
$176$	0	0
$177$	−23.7063	−1.78188
$178$	0	0
$179$	−2.38896	−0.178559	−0.0892797	−	0.996007i	$-0.528457\pi$
$179$	−2.38896	−0.178559	−0.0892797	+	0.996007i	$0.528457\pi$
$180$	0	0
$181$	3.33020	0.247531	0.123766	−	0.992311i	$-0.460503\pi$
$181$	3.33020	0.247531	0.123766	+	0.992311i	$0.460503\pi$
$182$	0	0
$183$	5.35136	0.395584
$184$	0	0
$185$	−0.914980	−0.0672706
$186$	0	0
$187$	1.30252	0.0952497
$188$	0	0
$189$	−23.5468	−1.71278
$190$	0	0
$191$	−17.9414	−1.29819	−0.649097	−	0.760705i	$-0.724853\pi$
$191$	−17.9414	−1.29819	−0.649097	+	0.760705i	$0.724853\pi$
$192$	0	0
$193$	13.5182	0.973060	0.486530	−	0.873664i	$-0.338262\pi$
$193$	13.5182	0.973060	0.486530	+	0.873664i	$0.338262\pi$
$194$	0	0
$195$	0.634674	0.0454499
$196$	0	0
$197$	14.7897	1.05372	0.526862	−	0.849951i	$-0.323368\pi$
$197$	14.7897	1.05372	0.526862	+	0.849951i	$0.323368\pi$
$198$	0	0
$199$	−22.8221	−1.61782	−0.808909	−	0.587934i	$-0.799941\pi$
$199$	−22.8221	−1.61782	−0.808909	+	0.587934i	$0.799941\pi$
$200$	0	0
$201$	13.0088	0.917573
$202$	0	0
$203$	4.99779	0.350776
$204$	0	0
$205$	2.14101	0.149534
$206$	0	0
$207$	1.28565	0.0893586
$208$	0	0
$209$	−25.6780	−1.77618
$210$	0	0
$211$	21.5893	1.48627	0.743136	−	0.669140i	$-0.233338\pi$
$211$	21.5893	1.48627	0.743136	+	0.669140i	$0.233338\pi$
$212$	0	0
$213$	−12.9070	−0.884374
$214$	0	0
$215$	−0.790036	−0.0538800
$216$	0	0
$217$	50.5625	3.43241
$218$	0	0
$219$	−7.31219	−0.494112
$220$	0	0
$221$	−0.561627	−0.0377791
$222$	0	0
$223$	−17.1114	−1.14586	−0.572931	−	0.819604i	$-0.694194\pi$
$223$	−17.1114	−1.14586	−0.572931	+	0.819604i	$0.694194\pi$
$224$	0	0
$225$	−2.33723	−0.155815
$226$	0	0
$227$	−16.7130	−1.10928	−0.554639	−	0.832091i	$-0.687144\pi$
$227$	−16.7130	−1.10928	−0.554639	+	0.832091i	$0.687144\pi$
$228$	0	0
$229$	−1.92544	−0.127236	−0.0636182	−	0.997974i	$-0.520264\pi$
$229$	−1.92544	−0.127236	−0.0636182	+	0.997974i	$0.520264\pi$
$230$	0	0
$231$	−36.8472	−2.42437
$232$	0	0
$233$	−9.96014	−0.652511	−0.326255	−	0.945282i	$-0.605787\pi$
$233$	−9.96014	−0.652511	−0.326255	+	0.945282i	$0.605787\pi$
$234$	0	0
$235$	−0.190657	−0.0124371
$236$	0	0
$237$	−12.1056	−0.786341
$238$	0	0
$239$	−6.54030	−0.423057	−0.211528	−	0.977372i	$-0.567844\pi$
$239$	−6.54030	−0.423057	−0.211528	+	0.977372i	$0.567844\pi$
$240$	0	0
$241$	−9.37270	−0.603749	−0.301874	−	0.953348i	$-0.597612\pi$
$241$	−9.37270	−0.603749	−0.301874	+	0.953348i	$0.597612\pi$
$242$	0	0
$243$	−4.85375	−0.311368
$244$	0	0
$245$	3.58921	0.229306
$246$	0	0
$247$	11.0720	0.704492
$248$	0	0
$249$	−22.8304	−1.44682
$250$	0	0
$251$	−14.0971	−0.889799	−0.444900	−	0.895580i	$-0.646761\pi$
$251$	−14.0971	−0.889799	−0.444900	+	0.895580i	$0.646761\pi$
$252$	0	0
$253$	−10.7970	−0.678800
$254$	0	0
$255$	−0.122433	−0.00766704
$256$	0	0
$257$	7.50668	0.468254	0.234127	−	0.972206i	$-0.424777\pi$
$257$	7.50668	0.468254	0.234127	+	0.972206i	$0.424777\pi$
$258$	0	0
$259$	−22.9049	−1.42324
$260$	0	0
$261$	0.471201	0.0291666
$262$	0	0
$263$	25.2410	1.55643	0.778214	−	0.627999i	$-0.216126\pi$
$263$	25.2410	1.55643	0.778214	+	0.627999i	$0.216126\pi$
$264$	0	0
$265$	0.457932	0.0281305
$266$	0	0
$267$	27.7831	1.70030
$268$	0	0
$269$	19.6310	1.19693	0.598463	−	0.801151i	$-0.295779\pi$
$269$	19.6310	1.19693	0.598463	+	0.801151i	$0.295779\pi$
$270$	0	0
$271$	−19.4336	−1.18051	−0.590253	−	0.807218i	$-0.700972\pi$
$271$	−19.4336	−1.18051	−0.590253	+	0.807218i	$0.700972\pi$
$272$	0	0
$273$	15.8880	0.961583
$274$	0	0
$275$	19.6282	1.18363
$276$	0	0
$277$	−17.7035	−1.06370	−0.531850	−	0.846838i	$-0.678503\pi$
$277$	−17.7035	−1.06370	−0.531850	+	0.846838i	$0.678503\pi$
$278$	0	0
$279$	4.76713	0.285401
$280$	0	0
$281$	−1.16156	−0.0692927	−0.0346464	−	0.999400i	$-0.511030\pi$
$281$	−1.16156	−0.0692927	−0.0346464	+	0.999400i	$0.511030\pi$
$282$	0	0
$283$	16.0569	0.954482	0.477241	−	0.878773i	$-0.341637\pi$
$283$	16.0569	0.954482	0.477241	+	0.878773i	$0.341637\pi$
$284$	0	0
$285$	2.41365	0.142972
$286$	0	0
$287$	53.5964	3.16369
$288$	0	0
$289$	−16.8917	−0.993627
$290$	0	0
$291$	6.97125	0.408662
$292$	0	0
$293$	16.7224	0.976934	0.488467	−	0.872583i	$-0.337556\pi$
$293$	16.7224	0.976934	0.488467	+	0.872583i	$0.337556\pi$
$294$	0	0
$295$	−2.54030	−0.147902
$296$	0	0
$297$	18.6441	1.08184
$298$	0	0
$299$	4.65549	0.269234
$300$	0	0
$301$	−19.7772	−1.13994
$302$	0	0
$303$	−19.3586	−1.11212
$304$	0	0
$305$	0.573436	0.0328348
$306$	0	0
$307$	−9.25690	−0.528319	−0.264159	−	0.964479i	$-0.585095\pi$
$307$	−9.25690	−0.528319	−0.264159	+	0.964479i	$0.585095\pi$
$308$	0	0
$309$	−13.1684	−0.749122
$310$	0	0
$311$	−25.9414	−1.47100	−0.735501	−	0.677524i	$-0.763053\pi$
$311$	−25.9414	−1.47100	−0.735501	+	0.677524i	$0.763053\pi$
$312$	0	0
$313$	−14.7311	−0.832651	−0.416325	−	0.909216i	$-0.636682\pi$
$313$	−14.7311	−0.832651	−0.416325	+	0.909216i	$0.636682\pi$
$314$	0	0
$315$	0.470159	0.0264904
$316$	0	0
$317$	−1.95728	−0.109932	−0.0549660	−	0.998488i	$-0.517505\pi$
$317$	−1.95728	−0.109932	−0.0549660	+	0.998488i	$0.517505\pi$
$318$	0	0
$319$	−3.95719	−0.221560
$320$	0	0
$321$	−16.4134	−0.916106
$322$	0	0
$323$	−2.13585	−0.118842
$324$	0	0
$325$	−8.46339	−0.469465
$326$	0	0
$327$	24.9018	1.37707
$328$	0	0
$329$	−4.77277	−0.263131
$330$	0	0
$331$	7.96122	0.437588	0.218794	−	0.975771i	$-0.429788\pi$
$331$	7.96122	0.437588	0.218794	+	0.975771i	$0.429788\pi$
$332$	0	0
$333$	−2.15952	−0.118341
$334$	0	0
$335$	1.39399	0.0761617
$336$	0	0
$337$	−18.4819	−1.00677	−0.503387	−	0.864061i	$-0.667913\pi$
$337$	−18.4819	−1.00677	−0.503387	+	0.864061i	$0.667913\pi$
$338$	0	0
$339$	−1.88784	−0.102533
$340$	0	0
$341$	−40.0348	−2.16801
$342$	0	0
$343$	54.8651	2.96244
$344$	0	0
$345$	1.01488	0.0546394
$346$	0	0
$347$	−4.01387	−0.215476	−0.107738	−	0.994179i	$-0.534361\pi$
$347$	−4.01387	−0.215476	−0.107738	+	0.994179i	$0.534361\pi$
$348$	0	0
$349$	17.6744	0.946091	0.473045	−	0.881038i	$-0.343155\pi$
$349$	17.6744	0.946091	0.473045	+	0.881038i	$0.343155\pi$
$350$	0	0
$351$	−8.03905	−0.429093
$352$	0	0
$353$	−34.5500	−1.83891	−0.919455	−	0.393196i	$-0.871369\pi$
$353$	−34.5500	−1.83891	−0.919455	+	0.393196i	$0.871369\pi$
$354$	0	0
$355$	−1.38308	−0.0734061
$356$	0	0
$357$	−3.06489	−0.162211
$358$	0	0
$359$	−1.21990	−0.0643836	−0.0321918	−	0.999482i	$-0.510249\pi$
$359$	−1.21990	−0.0643836	−0.0321918	+	0.999482i	$0.510249\pi$
$360$	0	0
$361$	23.1063	1.21612
$362$	0	0
$363$	8.68091	0.455629
$364$	0	0
$365$	−0.783552	−0.0410130
$366$	0	0
$367$	−12.8608	−0.671329	−0.335664	−	0.941982i	$-0.608961\pi$
$367$	−12.8608	−0.671329	−0.335664	+	0.941982i	$0.608961\pi$
$368$	0	0
$369$	5.05317	0.263058
$370$	0	0
$371$	11.4635	0.595156
$372$	0	0
$373$	14.5171	0.751669	0.375834	−	0.926687i	$-0.377356\pi$
$373$	14.5171	0.751669	0.375834	+	0.926687i	$0.377356\pi$
$374$	0	0
$375$	−3.70481	−0.191315
$376$	0	0
$377$	1.70628	0.0878779
$378$	0	0
$379$	14.3177	0.735451	0.367725	−	0.929934i	$-0.380137\pi$
$379$	14.3177	0.735451	0.367725	+	0.929934i	$0.380137\pi$
$380$	0	0
$381$	6.88381	0.352668
$382$	0	0
$383$	19.5831	1.00065	0.500326	−	0.865837i	$-0.333214\pi$
$383$	19.5831	1.00065	0.500326	+	0.865837i	$0.333214\pi$
$384$	0	0
$385$	−3.94843	−0.201231
$386$	0	0
$387$	−1.86463	−0.0947846
$388$	0	0
$389$	−23.7277	−1.20304	−0.601521	−	0.798857i	$-0.705439\pi$
$389$	−23.7277	−1.20304	−0.601521	+	0.798857i	$0.705439\pi$
$390$	0	0
$391$	−0.898075	−0.0454176
$392$	0	0
$393$	−7.75977	−0.391428
$394$	0	0
$395$	−1.29720	−0.0652690
$396$	0	0
$397$	−25.1687	−1.26318	−0.631590	−	0.775303i	$-0.717597\pi$
$397$	−25.1687	−1.26318	−0.631590	+	0.775303i	$0.717597\pi$
$398$	0	0
$399$	60.4215	3.02486
$400$	0	0
$401$	−21.3441	−1.06587	−0.532936	−	0.846155i	$-0.678911\pi$
$401$	−21.3441	−1.06587	−0.532936	+	0.846155i	$0.678911\pi$
$402$	0	0
$403$	17.2624	0.859902
$404$	0	0
$405$	−2.03471	−0.101105
$406$	0	0
$407$	18.1359	0.898961
$408$	0	0
$409$	22.3674	1.10600	0.552999	−	0.833182i	$-0.313483\pi$
$409$	22.3674	1.10600	0.552999	+	0.833182i	$0.313483\pi$
$410$	0	0
$411$	−21.6052	−1.06571
$412$	0	0
$413$	−63.5919	−3.12916
$414$	0	0
$415$	−2.44643	−0.120091
$416$	0	0
$417$	8.33367	0.408101
$418$	0	0
$419$	−30.6312	−1.49643	−0.748215	−	0.663456i	$-0.769089\pi$
$419$	−30.6312	−1.49643	−0.748215	+	0.663456i	$0.769089\pi$
$420$	0	0
$421$	16.8111	0.819324	0.409662	−	0.912237i	$-0.365647\pi$
$421$	16.8111	0.819324	0.409662	+	0.912237i	$0.365647\pi$
$422$	0	0
$423$	−0.449986	−0.0218791
$424$	0	0
$425$	1.63264	0.0791949
$426$	0	0
$427$	14.3550	0.694685
$428$	0	0
$429$	−12.5799	−0.607363
$430$	0	0
$431$	21.8731	1.05359	0.526796	−	0.849992i	$-0.323393\pi$
$431$	21.8731	1.05359	0.526796	+	0.849992i	$0.323393\pi$
$432$	0	0
$433$	17.6176	0.846648	0.423324	−	0.905978i	$-0.360863\pi$
$433$	17.6176	0.846648	0.423324	+	0.905978i	$0.360863\pi$
$434$	0	0
$435$	0.371963	0.0178343
$436$	0	0
$437$	17.7047	0.846931
$438$	0	0
$439$	−34.7994	−1.66088	−0.830442	−	0.557105i	$-0.811912\pi$
$439$	−34.7994	−1.66088	−0.830442	+	0.557105i	$0.811912\pi$
$440$	0	0
$441$	8.47120	0.403391
$442$	0	0
$443$	8.15463	0.387438	0.193719	−	0.981057i	$-0.437945\pi$
$443$	8.15463	0.387438	0.193719	+	0.981057i	$0.437945\pi$
$444$	0	0
$445$	2.97715	0.141131
$446$	0	0
$447$	38.7516	1.83289
$448$	0	0
$449$	−11.3748	−0.536812	−0.268406	−	0.963306i	$-0.586497\pi$
$449$	−11.3748	−0.536812	−0.268406	+	0.963306i	$0.586497\pi$
$450$	0	0
$451$	−42.4370	−1.99828
$452$	0	0
$453$	30.4259	1.42953
$454$	0	0
$455$	1.70251	0.0798147
$456$	0	0
$457$	−33.2334	−1.55459	−0.777295	−	0.629136i	$-0.783409\pi$
$457$	−33.2334	−1.55459	−0.777295	+	0.629136i	$0.783409\pi$
$458$	0	0
$459$	1.55079	0.0723845
$460$	0	0
$461$	24.8740	1.15850	0.579248	−	0.815151i	$-0.303346\pi$
$461$	24.8740	1.15850	0.579248	+	0.815151i	$0.303346\pi$
$462$	0	0
$463$	−10.2893	−0.478182	−0.239091	−	0.970997i	$-0.576849\pi$
$463$	−10.2893	−0.478182	−0.239091	+	0.970997i	$0.576849\pi$
$464$	0	0
$465$	3.76314	0.174512
$466$	0	0
$467$	31.3152	1.44909	0.724547	−	0.689225i	$-0.242049\pi$
$467$	31.3152	1.44909	0.724547	+	0.689225i	$0.242049\pi$
$468$	0	0
$469$	34.8961	1.61135
$470$	0	0
$471$	−33.8649	−1.56041
$472$	0	0
$473$	15.6593	0.720018
$474$	0	0
$475$	−32.1861	−1.47680
$476$	0	0
$477$	1.08080	0.0494866
$478$	0	0
$479$	30.0623	1.37358	0.686791	−	0.726855i	$-0.259019\pi$
$479$	30.0623	1.37358	0.686791	+	0.726855i	$0.259019\pi$
$480$	0	0
$481$	−7.81991	−0.356557
$482$	0	0
$483$	25.4058	1.15600
$484$	0	0
$485$	0.747018	0.0339203
$486$	0	0
$487$	−0.690021	−0.0312678	−0.0156339	−	0.999878i	$-0.504977\pi$
$487$	−0.690021	−0.0312678	−0.0156339	+	0.999878i	$0.504977\pi$
$488$	0	0
$489$	−26.4240	−1.19493
$490$	0	0
$491$	−31.4864	−1.42096	−0.710481	−	0.703716i	$-0.751523\pi$
$491$	−31.4864	−1.42096	−0.710481	+	0.703716i	$0.751523\pi$
$492$	0	0
$493$	−0.329153	−0.0148243
$494$	0	0
$495$	−0.372266	−0.0167321
$496$	0	0
$497$	−34.6229	−1.55305
$498$	0	0
$499$	−10.7970	−0.483339	−0.241669	−	0.970359i	$-0.577695\pi$
$499$	−10.7970	−0.483339	−0.241669	+	0.970359i	$0.577695\pi$
$500$	0	0
$501$	29.2998	1.30902
$502$	0	0
$503$	−12.8338	−0.572229	−0.286114	−	0.958195i	$-0.592364\pi$
$503$	−12.8338	−0.572229	−0.286114	+	0.958195i	$0.592364\pi$
$504$	0	0
$505$	−2.07441	−0.0923100
$506$	0	0
$507$	−18.7962	−0.834770
$508$	0	0
$509$	−15.0126	−0.665421	−0.332711	−	0.943029i	$-0.607963\pi$
$509$	−15.0126	−0.665421	−0.332711	+	0.943029i	$0.607963\pi$
$510$	0	0
$511$	−19.6149	−0.867710
$512$	0	0
$513$	−30.5723	−1.34980
$514$	0	0
$515$	−1.41108	−0.0621797
$516$	0	0
$517$	3.77902	0.166201
$518$	0	0
$519$	−8.46434	−0.371543
$520$	0	0
$521$	−33.4337	−1.46476	−0.732379	−	0.680897i	$-0.761590\pi$
$521$	−33.4337	−1.46476	−0.732379	+	0.680897i	$0.761590\pi$
$522$	0	0
$523$	−11.1874	−0.489189	−0.244594	−	0.969625i	$-0.578655\pi$
$523$	−11.1874	−0.489189	−0.244594	+	0.969625i	$0.578655\pi$
$524$	0	0
$525$	−46.1862	−2.01573
$526$	0	0
$527$	−3.33003	−0.145058
$528$	0	0
$529$	−15.5556	−0.676330
$530$	0	0
$531$	−5.99558	−0.260186
$532$	0	0
$533$	18.2982	0.792583
$534$	0	0
$535$	−1.75881	−0.0760399
$536$	0	0
$537$	−4.45091	−0.192071
$538$	0	0
$539$	−71.1419	−3.06430
$540$	0	0
$541$	5.56974	0.239462	0.119731	−	0.992806i	$-0.461797\pi$
$541$	5.56974	0.239462	0.119731	+	0.992806i	$0.461797\pi$
$542$	0	0
$543$	6.20454	0.266262
$544$	0	0
$545$	2.66840	0.114302
$546$	0	0
$547$	−40.2326	−1.72022	−0.860111	−	0.510107i	$-0.829606\pi$
$547$	−40.2326	−1.72022	−0.860111	+	0.510107i	$0.829606\pi$
$548$	0	0
$549$	1.35341	0.0577623
$550$	0	0
$551$	6.48894	0.276438
$552$	0	0
$553$	−32.4731	−1.38089
$554$	0	0
$555$	−1.70471	−0.0723610
$556$	0	0
$557$	2.17930	0.0923398	0.0461699	−	0.998934i	$-0.485298\pi$
$557$	2.17930	0.0923398	0.0461699	+	0.998934i	$0.485298\pi$
$558$	0	0
$559$	−6.75208	−0.285582
$560$	0	0
$561$	2.42675	0.102457
$562$	0	0
$563$	−25.8852	−1.09093	−0.545466	−	0.838133i	$-0.683647\pi$
$563$	−25.8852	−1.09093	−0.545466	+	0.838133i	$0.683647\pi$
$564$	0	0
$565$	−0.202295	−0.00851062
$566$	0	0
$567$	−50.9353	−2.13908
$568$	0	0
$569$	24.1837	1.01383	0.506917	−	0.861995i	$-0.330785\pi$
$569$	24.1837	1.01383	0.506917	+	0.861995i	$0.330785\pi$
$570$	0	0
$571$	4.66716	0.195314	0.0976572	−	0.995220i	$-0.468865\pi$
$571$	4.66716	0.195314	0.0976572	+	0.995220i	$0.468865\pi$
$572$	0	0
$573$	−33.4269	−1.39643
$574$	0	0
$575$	−13.5335	−0.564385
$576$	0	0
$577$	39.5832	1.64787	0.823934	−	0.566685i	$-0.191774\pi$
$577$	39.5832	1.64787	0.823934	+	0.566685i	$0.191774\pi$
$578$	0	0
$579$	25.1859	1.04669
$580$	0	0
$581$	−61.2422	−2.54075
$582$	0	0
$583$	−9.07668	−0.375918
$584$	0	0
$585$	0.160516	0.00663650
$586$	0	0
$587$	28.0569	1.15803	0.579015	−	0.815317i	$-0.303437\pi$
$587$	28.0569	1.15803	0.579015	+	0.815317i	$0.303437\pi$
$588$	0	0
$589$	65.6485	2.70500
$590$	0	0
$591$	27.5550	1.13346
$592$	0	0
$593$	0.484471	0.0198948	0.00994742	−	0.999951i	$-0.496834\pi$
$593$	0.484471	0.0198948	0.00994742	+	0.999951i	$0.496834\pi$
$594$	0	0
$595$	−0.328425	−0.0134641
$596$	0	0
$597$	−42.5203	−1.74024
$598$	0	0
$599$	−34.5980	−1.41363	−0.706817	−	0.707396i	$-0.749870\pi$
$599$	−34.5980	−1.41363	−0.706817	+	0.707396i	$0.749870\pi$
$600$	0	0
$601$	−2.53587	−0.103440	−0.0517202	−	0.998662i	$-0.516470\pi$
$601$	−2.53587	−0.103440	−0.0517202	+	0.998662i	$0.516470\pi$
$602$	0	0
$603$	3.29007	0.133982
$604$	0	0
$605$	0.930220	0.0378188
$606$	0	0
$607$	−25.9200	−1.05206	−0.526030	−	0.850466i	$-0.676320\pi$
$607$	−25.9200	−1.05206	−0.526030	+	0.850466i	$0.676320\pi$
$608$	0	0
$609$	9.31146	0.377319
$610$	0	0
$611$	−1.62946	−0.0659208
$612$	0	0
$613$	−29.6158	−1.19617	−0.598086	−	0.801432i	$-0.704072\pi$
$613$	−29.6158	−1.19617	−0.598086	+	0.801432i	$0.704072\pi$
$614$	0	0
$615$	3.98894	0.160850
$616$	0	0
$617$	−6.88636	−0.277234	−0.138617	−	0.990346i	$-0.544266\pi$
$617$	−6.88636	−0.277234	−0.138617	+	0.990346i	$0.544266\pi$
$618$	0	0
$619$	−30.5755	−1.22893	−0.614466	−	0.788943i	$-0.710628\pi$
$619$	−30.5755	−1.22893	−0.614466	+	0.788943i	$0.710628\pi$
$620$	0	0
$621$	−12.8549	−0.515850
$622$	0	0
$623$	74.5278	2.98589
$624$	0	0
$625$	24.4037	0.976148
$626$	0	0
$627$	−47.8410	−1.91059
$628$	0	0
$629$	1.50851	0.0601483
$630$	0	0
$631$	13.5337	0.538767	0.269383	−	0.963033i	$-0.413180\pi$
$631$	13.5337	0.538767	0.269383	+	0.963033i	$0.413180\pi$
$632$	0	0
$633$	40.2235	1.59874
$634$	0	0
$635$	0.737648	0.0292727
$636$	0	0
$637$	30.6753	1.21540
$638$	0	0
$639$	−3.26432	−0.129134
$640$	0	0
$641$	47.7330	1.88534	0.942669	−	0.333729i	$-0.108307\pi$
$641$	47.7330	1.88534	0.942669	+	0.333729i	$0.108307\pi$
$642$	0	0
$643$	−1.89024	−0.0745436	−0.0372718	−	0.999305i	$-0.511867\pi$
$643$	−1.89024	−0.0745436	−0.0372718	+	0.999305i	$0.511867\pi$
$644$	0	0
$645$	−1.47193	−0.0579572
$646$	0	0
$647$	−33.4884	−1.31657	−0.658283	−	0.752771i	$-0.728717\pi$
$647$	−33.4884	−1.31657	−0.658283	+	0.752771i	$0.728717\pi$
$648$	0	0
$649$	50.3514	1.97646
$650$	0	0
$651$	94.2038	3.69214
$652$	0	0
$653$	8.87678	0.347375	0.173688	−	0.984801i	$-0.444432\pi$
$653$	8.87678	0.347375	0.173688	+	0.984801i	$0.444432\pi$
$654$	0	0
$655$	−0.831513	−0.0324899
$656$	0	0
$657$	−1.84933	−0.0721491
$658$	0	0
$659$	23.6963	0.923077	0.461538	−	0.887120i	$-0.347298\pi$
$659$	23.6963	0.923077	0.461538	+	0.887120i	$0.347298\pi$
$660$	0	0
$661$	8.95134	0.348167	0.174083	−	0.984731i	$-0.444304\pi$
$661$	8.95134	0.348167	0.174083	+	0.984731i	$0.444304\pi$
$662$	0	0
$663$	−1.04638	−0.0406379
$664$	0	0
$665$	6.47458	0.251074
$666$	0	0
$667$	2.72844	0.105646
$668$	0	0
$669$	−31.8805	−1.23257
$670$	0	0
$671$	−11.3661	−0.438783
$672$	0	0
$673$	49.1527	1.89470	0.947349	−	0.320203i	$-0.103751\pi$
$673$	49.1527	1.89470	0.947349	+	0.320203i	$0.103751\pi$
$674$	0	0
$675$	23.3694	0.899490
$676$	0	0
$677$	30.8288	1.18485	0.592423	−	0.805627i	$-0.298172\pi$
$677$	30.8288	1.18485	0.592423	+	0.805627i	$0.298172\pi$
$678$	0	0
$679$	18.7003	0.717651
$680$	0	0
$681$	−31.1382	−1.19322
$682$	0	0
$683$	−15.1226	−0.578650	−0.289325	−	0.957231i	$-0.593431\pi$
$683$	−15.1226	−0.578650	−0.289325	+	0.957231i	$0.593431\pi$
$684$	0	0
$685$	−2.31515	−0.0884573
$686$	0	0
$687$	−3.58731	−0.136864
$688$	0	0
$689$	3.91373	0.149101
$690$	0	0
$691$	4.17367	0.158774	0.0793870	−	0.996844i	$-0.474704\pi$
$691$	4.17367	0.158774	0.0793870	+	0.996844i	$0.474704\pi$
$692$	0	0
$693$	−9.31904	−0.354001
$694$	0	0
$695$	0.893010	0.0338738
$696$	0	0
$697$	−3.52984	−0.133702
$698$	0	0
$699$	−18.5569	−0.701886
$700$	0	0
$701$	2.22431	0.0840110	0.0420055	−	0.999117i	$-0.486625\pi$
$701$	2.22431	0.0840110	0.0420055	+	0.999117i	$0.486625\pi$
$702$	0	0
$703$	−29.7389	−1.12162
$704$	0	0
$705$	−0.355216	−0.0133782
$706$	0	0
$707$	−51.9292	−1.95300
$708$	0	0
$709$	27.9210	1.04859	0.524297	−	0.851535i	$-0.324328\pi$
$709$	27.9210	1.04859	0.524297	+	0.851535i	$0.324328\pi$
$710$	0	0
$711$	−3.06162	−0.114820
$712$	0	0
$713$	27.6036	1.03376
$714$	0	0
$715$	−1.34802	−0.0504132
$716$	0	0
$717$	−12.1853	−0.455070
$718$	0	0
$719$	13.7594	0.513138	0.256569	−	0.966526i	$-0.417408\pi$
$719$	13.7594	0.513138	0.256569	+	0.966526i	$0.417408\pi$
$720$	0	0
$721$	−35.3240	−1.31553
$722$	0	0
$723$	−17.4624	−0.649435
$724$	0	0
$725$	−4.96014	−0.184215
$726$	0	0
$727$	27.8685	1.03358	0.516792	−	0.856111i	$-0.327126\pi$
$727$	27.8685	1.03358	0.516792	+	0.856111i	$0.327126\pi$
$728$	0	0
$729$	21.5316	0.797468
$730$	0	0
$731$	1.30252	0.0481754
$732$	0	0
$733$	23.0576	0.851652	0.425826	−	0.904805i	$-0.359984\pi$
$733$	23.0576	0.851652	0.425826	+	0.904805i	$0.359984\pi$
$734$	0	0
$735$	6.68711	0.246658
$736$	0	0
$737$	−27.6303	−1.01778
$738$	0	0
$739$	44.5880	1.64019	0.820097	−	0.572224i	$-0.193919\pi$
$739$	44.5880	1.64019	0.820097	+	0.572224i	$0.193919\pi$
$740$	0	0
$741$	20.6283	0.757801
$742$	0	0
$743$	33.0232	1.21150	0.605752	−	0.795654i	$-0.292872\pi$
$743$	33.0232	1.21150	0.605752	+	0.795654i	$0.292872\pi$
$744$	0	0
$745$	4.15251	0.152136
$746$	0	0
$747$	−5.77404	−0.211261
$748$	0	0
$749$	−44.0287	−1.60877
$750$	0	0
$751$	7.39829	0.269968	0.134984	−	0.990848i	$-0.456902\pi$
$751$	7.39829	0.269968	0.134984	+	0.990848i	$0.456902\pi$
$752$	0	0
$753$	−26.2645	−0.957131
$754$	0	0
$755$	3.26034	0.118656
$756$	0	0
$757$	−28.0125	−1.01813	−0.509066	−	0.860727i	$-0.670009\pi$
$757$	−28.0125	−1.01813	−0.509066	+	0.860727i	$0.670009\pi$
$758$	0	0
$759$	−20.1160	−0.730165
$760$	0	0
$761$	21.7514	0.788487	0.394243	−	0.919006i	$-0.371007\pi$
$761$	21.7514	0.788487	0.394243	+	0.919006i	$0.371007\pi$
$762$	0	0
$763$	66.7988	2.41828
$764$	0	0
$765$	−0.0309645	−0.00111952
$766$	0	0
$767$	−21.7107	−0.783930
$768$	0	0
$769$	−41.4568	−1.49497	−0.747486	−	0.664278i	$-0.768739\pi$
$769$	−41.4568	−1.49497	−0.747486	+	0.664278i	$0.768739\pi$
$770$	0	0
$771$	13.9858	0.503687
$772$	0	0
$773$	−16.3897	−0.589497	−0.294749	−	0.955575i	$-0.595236\pi$
$773$	−16.3897	−0.589497	−0.294749	+	0.955575i	$0.595236\pi$
$774$	0	0
$775$	−50.1816	−1.80258
$776$	0	0
$777$	−42.6745	−1.53094
$778$	0	0
$779$	69.5875	2.49323
$780$	0	0
$781$	27.4140	0.980951
$782$	0	0
$783$	−4.71145	−0.168373
$784$	0	0
$785$	−3.62886	−0.129520
$786$	0	0
$787$	3.51597	0.125331	0.0626654	−	0.998035i	$-0.480040\pi$
$787$	3.51597	0.125331	0.0626654	+	0.998035i	$0.480040\pi$
$788$	0	0
$789$	47.0269	1.67420
$790$	0	0
$791$	−5.06411	−0.180059
$792$	0	0
$793$	4.90089	0.174036
$794$	0	0
$795$	0.853180	0.0302592
$796$	0	0
$797$	20.1889	0.715127	0.357563	−	0.933889i	$-0.383608\pi$
$797$	20.1889	0.715127	0.357563	+	0.933889i	$0.383608\pi$
$798$	0	0
$799$	0.314333	0.0111203
$800$	0	0
$801$	7.02663	0.248274
$802$	0	0
$803$	15.5308	0.548070
$804$	0	0
$805$	2.72241	0.0959523
$806$	0	0
$807$	36.5749	1.28750
$808$	0	0
$809$	10.4162	0.366214	0.183107	−	0.983093i	$-0.441385\pi$
$809$	10.4162	0.366214	0.183107	+	0.983093i	$0.441385\pi$
$810$	0	0
$811$	41.6472	1.46243	0.731216	−	0.682146i	$-0.238953\pi$
$811$	41.6472	1.46243	0.731216	+	0.682146i	$0.238953\pi$
$812$	0	0
$813$	−36.2070	−1.26984
$814$	0	0
$815$	−2.83152	−0.0991837
$816$	0	0
$817$	−25.6780	−0.898358
$818$	0	0
$819$	4.01823	0.140408
$820$	0	0
$821$	22.5899	0.788392	0.394196	−	0.919026i	$-0.371023\pi$
$821$	22.5899	0.788392	0.394196	+	0.919026i	$0.371023\pi$
$822$	0	0
$823$	20.2588	0.706178	0.353089	−	0.935590i	$-0.385131\pi$
$823$	20.2588	0.706178	0.353089	+	0.935590i	$0.385131\pi$
$824$	0	0
$825$	36.5697	1.27319
$826$	0	0
$827$	−6.59880	−0.229463	−0.114731	−	0.993397i	$-0.536601\pi$
$827$	−6.59880	−0.229463	−0.114731	+	0.993397i	$0.536601\pi$
$828$	0	0
$829$	11.6230	0.403683	0.201841	−	0.979418i	$-0.435307\pi$
$829$	11.6230	0.403683	0.201841	+	0.979418i	$0.435307\pi$
$830$	0	0
$831$	−32.9837	−1.14419
$832$	0	0
$833$	−5.91747	−0.205028
$834$	0	0
$835$	3.13968	0.108653
$836$	0	0
$837$	−47.6656	−1.64756
$838$	0	0
$839$	10.9047	0.376474	0.188237	−	0.982124i	$-0.439723\pi$
$839$	10.9047	0.376474	0.188237	+	0.982124i	$0.439723\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.16412	−0.0745362
$844$	0	0
$845$	−2.01415	−0.0692888
$846$	0	0
$847$	23.2864	0.800131
$848$	0	0
$849$	29.9158	1.02671
$850$	0	0
$851$	−12.5045	−0.428649
$852$	0	0
$853$	57.1847	1.95797	0.978983	−	0.203941i	$-0.0653750\pi$
$853$	57.1847	1.95797	0.978983	+	0.203941i	$0.0653750\pi$
$854$	0	0
$855$	0.610437	0.0208765
$856$	0	0
$857$	−16.4278	−0.561161	−0.280581	−	0.959830i	$-0.590527\pi$
$857$	−16.4278	−0.561161	−0.280581	+	0.959830i	$0.590527\pi$
$858$	0	0
$859$	2.29004	0.0781351	0.0390676	−	0.999237i	$-0.487561\pi$
$859$	2.29004	0.0781351	0.0390676	+	0.999237i	$0.487561\pi$
$860$	0	0
$861$	99.8563	3.40309
$862$	0	0
$863$	−11.6751	−0.397426	−0.198713	−	0.980058i	$-0.563676\pi$
$863$	−11.6751	−0.397426	−0.198713	+	0.980058i	$0.563676\pi$
$864$	0	0
$865$	−0.907013	−0.0308394
$866$	0	0
$867$	−31.4711	−1.06882
$868$	0	0
$869$	25.7118	0.872213
$870$	0	0
$871$	11.9138	0.403683
$872$	0	0
$873$	1.76310	0.0596719
$874$	0	0
$875$	−9.93811	−0.335969
$876$	0	0
$877$	44.9506	1.51787	0.758936	−	0.651165i	$-0.225719\pi$
$877$	44.9506	1.51787	0.758936	+	0.651165i	$0.225719\pi$
$878$	0	0
$879$	31.1558	1.05086
$880$	0	0
$881$	−28.9904	−0.976710	−0.488355	−	0.872645i	$-0.662403\pi$
$881$	−28.9904	−0.976710	−0.488355	+	0.872645i	$0.662403\pi$
$882$	0	0
$883$	19.6126	0.660018	0.330009	−	0.943978i	$-0.392948\pi$
$883$	19.6126	0.660018	0.330009	+	0.943978i	$0.392948\pi$
$884$	0	0
$885$	−4.73287	−0.159094
$886$	0	0
$887$	−12.3775	−0.415595	−0.207797	−	0.978172i	$-0.566629\pi$
$887$	−12.3775	−0.415595	−0.207797	+	0.978172i	$0.566629\pi$
$888$	0	0
$889$	18.4657	0.619321
$890$	0	0
$891$	40.3300	1.35111
$892$	0	0
$893$	−6.19679	−0.207368
$894$	0	0
$895$	−0.476946	−0.0159426
$896$	0	0
$897$	8.67372	0.289607
$898$	0	0
$899$	10.1170	0.337420
$900$	0	0
$901$	−0.754984	−0.0251522
$902$	0	0
$903$	−36.8472	−1.22620
$904$	0	0
$905$	0.664860	0.0221007
$906$	0	0
$907$	−57.5000	−1.90925	−0.954627	−	0.297803i	$-0.903746\pi$
$907$	−57.5000	−1.90925	−0.954627	+	0.297803i	$0.903746\pi$
$908$	0	0
$909$	−4.89599	−0.162390
$910$	0	0
$911$	−54.6102	−1.80932	−0.904659	−	0.426137i	$-0.859874\pi$
$911$	−54.6102	−1.80932	−0.904659	+	0.426137i	$0.859874\pi$
$912$	0	0
$913$	48.4908	1.60481
$914$	0	0
$915$	1.06838	0.0353195
$916$	0	0
$917$	−20.8155	−0.687388
$918$	0	0
$919$	11.9534	0.394306	0.197153	−	0.980373i	$-0.436830\pi$
$919$	11.9534	0.394306	0.197153	+	0.980373i	$0.436830\pi$
$920$	0	0
$921$	−17.2467	−0.568297
$922$	0	0
$923$	−11.8205	−0.389077
$924$	0	0
$925$	22.7324	0.747437
$926$	0	0
$927$	−3.33042	−0.109385
$928$	0	0
$929$	40.8501	1.34025	0.670124	−	0.742249i	$-0.266241\pi$
$929$	40.8501	1.34025	0.670124	+	0.742249i	$0.266241\pi$
$930$	0	0
$931$	116.657	3.82329
$932$	0	0
$933$	−48.3318	−1.58231
$934$	0	0
$935$	0.260043	0.00850431
$936$	0	0
$937$	−34.0245	−1.11153	−0.555767	−	0.831338i	$-0.687575\pi$
$937$	−34.0245	−1.11153	−0.555767	+	0.831338i	$0.687575\pi$
$938$	0	0
$939$	−27.4457	−0.895658
$940$	0	0
$941$	6.32369	0.206146	0.103073	−	0.994674i	$-0.467132\pi$
$941$	6.32369	0.206146	0.103073	+	0.994674i	$0.467132\pi$
$942$	0	0
$943$	29.2599	0.952833
$944$	0	0
$945$	−4.70102	−0.152924
$946$	0	0
$947$	38.9787	1.26664	0.633319	−	0.773891i	$-0.281692\pi$
$947$	38.9787	1.26664	0.633319	+	0.773891i	$0.281692\pi$
$948$	0	0
$949$	−6.69665	−0.217383
$950$	0	0
$951$	−3.64665	−0.118251
$952$	0	0
$953$	17.0428	0.552071	0.276036	−	0.961147i	$-0.410979\pi$
$953$	17.0428	0.552071	0.276036	+	0.961147i	$0.410979\pi$
$954$	0	0
$955$	−3.58193	−0.115908
$956$	0	0
$957$	−7.37270	−0.238326
$958$	0	0
$959$	−57.9558	−1.87149
$960$	0	0
$961$	71.3532	2.30172
$962$	0	0
$963$	−4.15111	−0.133768
$964$	0	0
$965$	2.69885	0.0868790
$966$	0	0
$967$	−33.4571	−1.07591	−0.537953	−	0.842975i	$-0.680802\pi$
$967$	−33.4571	−1.07591	−0.537953	+	0.842975i	$0.680802\pi$
$968$	0	0
$969$	−3.97934	−0.127835
$970$	0	0
$971$	24.1082	0.773670	0.386835	−	0.922149i	$-0.373568\pi$
$971$	24.1082	0.773670	0.386835	+	0.922149i	$0.373568\pi$
$972$	0	0
$973$	22.3550	0.716667
$974$	0	0
$975$	−15.7683	−0.504989
$976$	0	0
$977$	10.7968	0.345419	0.172709	−	0.984973i	$-0.444748\pi$
$977$	10.7968	0.345419	0.172709	+	0.984973i	$0.444748\pi$
$978$	0	0
$979$	−59.0103	−1.88598
$980$	0	0
$981$	6.29792	0.201077
$982$	0	0
$983$	33.5906	1.07137	0.535686	−	0.844417i	$-0.320053\pi$
$983$	33.5906	1.07137	0.535686	+	0.844417i	$0.320053\pi$
$984$	0	0
$985$	2.95271	0.0940811
$986$	0	0
$987$	−8.89223	−0.283043
$988$	0	0
$989$	−10.7970	−0.343324
$990$	0	0
$991$	−11.2472	−0.357280	−0.178640	−	0.983915i	$-0.557170\pi$
$991$	−11.2472	−0.357280	−0.178640	+	0.983915i	$0.557170\pi$
$992$	0	0
$993$	14.8327	0.470701
$994$	0	0
$995$	−4.55634	−0.144446
$996$	0	0
$997$	−46.7100	−1.47932	−0.739661	−	0.672980i	$-0.765014\pi$
$997$	−46.7100	−1.47932	−0.739661	+	0.672980i	$0.765014\pi$
$998$	0	0
$999$	21.5926	0.683160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.ba.1.5		5
4.3	odd	2		1856.2.a.bb.1.1		5
8.3	odd	2		928.2.a.g.1.5	✓	5
8.5	even	2		928.2.a.h.1.1	yes	5
24.5	odd	2		8352.2.a.bh.1.3		5
24.11	even	2		8352.2.a.bg.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.g.1.5	✓	5	8.3	odd	2
928.2.a.h.1.1	yes	5	8.5	even	2
1856.2.a.ba.1.5		5	1.1	even	1	trivial
1856.2.a.bb.1.1		5	4.3	odd	2
8352.2.a.bg.1.3		5	24.11	even	2
8352.2.a.bh.1.3		5	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.86312 q^{3} +0.199646 q^{5} +4.99779 q^{7} +0.471201 q^{9} +O(q^{10})\)	\(q+1.86312 q^{3} +0.199646 q^{5} +4.99779 q^{7} +0.471201 q^{9} -3.95719 q^{11} +1.70628 q^{13} +0.371963 q^{15} -0.329153 q^{17} +6.48894 q^{19} +9.31146 q^{21} +2.72844 q^{23} -4.96014 q^{25} -4.71145 q^{27} +1.00000 q^{29} +10.1170 q^{31} -7.37270 q^{33} +0.997788 q^{35} -4.58301 q^{37} +3.17900 q^{39} +10.7240 q^{41} -3.95719 q^{43} +0.0940734 q^{45} -0.954977 q^{47} +17.9779 q^{49} -0.613250 q^{51} +2.29372 q^{53} -0.790036 q^{55} +12.0896 q^{57} -12.7240 q^{59} +2.87226 q^{61} +2.35496 q^{63} +0.340652 q^{65} +6.98231 q^{67} +5.08341 q^{69} -6.92765 q^{71} -3.92471 q^{73} -9.24132 q^{75} -19.7772 q^{77} -6.49749 q^{79} -10.1916 q^{81} -12.2539 q^{83} -0.0657140 q^{85} +1.86312 q^{87} +14.9122 q^{89} +8.52763 q^{91} +18.8491 q^{93} +1.29549 q^{95} +3.74171 q^{97} -1.86463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9	\( 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9} - 8 q^{11} + 6 q^{13} + 6 q^{15} + 6 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{23} + 7 q^{25} - 22 q^{27} + 5 q^{29} + 8 q^{31} - 16 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} + 28 q^{47} + 13 q^{49} - 24 q^{51} + 14 q^{53} + 10 q^{55} + 20 q^{57} - 16 q^{59} + 18 q^{61} + 20 q^{63} - 8 q^{65} + 28 q^{69} - 4 q^{71} + 2 q^{73} + 6 q^{75} + 4 q^{77} - 12 q^{79} + 9 q^{81} - 32 q^{83} + 32 q^{85} - 4 q^{87} + 30 q^{89} + 40 q^{91} - 4 q^{93} + 4 q^{95} + 6 q^{97} + 22 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9 - 8 * q^11 + 6 * q^13 + 6 * q^15 + 6 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^23 + 7 * q^25 - 22 * q^27 + 5 * q^29 + 8 * q^31 - 16 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 + 28 * q^47 + 13 * q^49 - 24 * q^51 + 14 * q^53 + 10 * q^55 + 20 * q^57 - 16 * q^59 + 18 * q^61 + 20 * q^63 - 8 * q^65 + 28 * q^69 - 4 * q^71 + 2 * q^73 + 6 * q^75 + 4 * q^77 - 12 * q^79 + 9 * q^81 - 32 * q^83 + 32 * q^85 - 4 * q^87 + 30 * q^89 + 40 * q^91 - 4 * q^93 + 4 * q^95 + 6 * q^97 + 22 * q^99

Introduction

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