LMFDB - Embedded newform 928.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,2,Mod(1,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, 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("928.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	928.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:	$7.41011730757$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.49889$ of defining polynomial
Character	$\chi$	$=$	928.1

$q$-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	$-0.680758\pi$
$3$	−1.86312	−1.07567	−0.537835	+	0.843050i	$0.680758\pi$
$4$	0	0
$5$	−0.199646	−0.0892843	−0.0446422	−	0.999003i	$-0.514215\pi$
$5$	−0.199646	−0.0892843	−0.0446422	+	0.999003i	$0.514215\pi$
$6$	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	$-0.296530\pi$
$11$	3.95719	1.19314	0.596569	+	0.802562i	$0.296530\pi$
$12$	0	0
$13$	−1.70628	−0.473237	−0.236619	−	0.971603i	$-0.576039\pi$
$13$	−1.70628	−0.473237	−0.236619	+	0.971603i	$0.576039\pi$
$14$	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	$-0.767232\pi$
$19$	−6.48894	−1.48866	−0.744332	+	0.667809i	$0.767232\pi$
$20$	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	$-0.377051\pi$
$37$	4.58301	0.753443	0.376721	+	0.926327i	$0.377051\pi$
$38$	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	$-0.402435\pi$
$43$	3.95719	0.603466	0.301733	+	0.953393i	$0.402435\pi$
$44$	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	$-0.550354\pi$
$53$	−2.29372	−0.315067	−0.157533	+	0.987514i	$0.550354\pi$
$54$	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	$-0.189329\pi$
$59$	12.7240	1.65653	0.828263	+	0.560339i	$0.189329\pi$
$60$	0	0
$61$	−2.87226	−0.367756	−0.183878	−	0.982949i	$-0.558865\pi$
$61$	−2.87226	−0.367756	−0.183878	+	0.982949i	$0.558865\pi$
$62$	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	$-0.640258\pi$
$67$	−6.98231	−0.853024	−0.426512	+	0.904482i	$0.640258\pi$
$68$	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	$-0.265213\pi$
$83$	12.2539	1.34504	0.672518	+	0.740081i	$0.265213\pi$
$84$	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	$-0.327070\pi$
$101$	10.3904	1.03389	0.516944	+	0.856019i	$0.327070\pi$
$102$	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	$-0.359982\pi$
$107$	8.80964	0.851660	0.425830	+	0.904803i	$0.359982\pi$
$108$	0	0
$109$	−13.3657	−1.28020	−0.640100	−	0.768292i	$-0.721107\pi$
$109$	−13.3657	−1.28020	−0.640100	+	0.768292i	$0.721107\pi$
$110$	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	$-0.441760\pi$
$131$	4.16494	0.363893	0.181946	+	0.983308i	$0.441760\pi$
$132$	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	$-0.560750\pi$
$139$	−4.47297	−0.379393	−0.189696	+	0.981843i	$0.560750\pi$
$140$	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	$-0.824595\pi$
$149$	−20.7994	−1.70395	−0.851975	+	0.523583i	$0.824595\pi$
$150$	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	$-0.241692\pi$
$157$	18.1765	1.45064	0.725321	+	0.688411i	$0.241692\pi$
$158$	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	$-0.312551\pi$
$163$	14.1827	1.11087	0.555437	+	0.831559i	$0.312551\pi$
$164$	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	$-0.444750\pi$
$173$	4.54311	0.345406	0.172703	+	0.984974i	$0.444750\pi$
$174$	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	$-0.471543\pi$
$179$	2.38896	0.178559	0.0892797	+	0.996007i	$0.471543\pi$
$180$	0	0
$181$	−3.33020	−0.247531	−0.123766	−	0.992311i	$-0.539497\pi$
$181$	−3.33020	−0.247531	−0.123766	+	0.992311i	$0.539497\pi$
$182$	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	$-0.676632\pi$
$197$	−14.7897	−1.05372	−0.526862	+	0.849951i	$0.676632\pi$
$198$	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	$-0.766662\pi$
$211$	−21.5893	−1.48627	−0.743136	+	0.669140i	$0.766662\pi$
$212$	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	$-0.312856\pi$
$227$	16.7130	1.10928	0.554639	+	0.832091i	$0.312856\pi$
$228$	0	0
$229$	1.92544	0.127236	0.0636182	−	0.997974i	$-0.479736\pi$
$229$	1.92544	0.127236	0.0636182	+	0.997974i	$0.479736\pi$
$230$	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	$-0.353239\pi$
$251$	14.0971	0.889799	0.444900	+	0.895580i	$0.353239\pi$
$252$	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	$-0.704221\pi$
$269$	−19.6310	−1.19693	−0.598463	+	0.801151i	$0.704221\pi$
$270$	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	$-0.321497\pi$
$277$	17.7035	1.06370	0.531850	+	0.846838i	$0.321497\pi$
$278$	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	$-0.658363\pi$
$283$	−16.0569	−0.954482	−0.477241	+	0.878773i	$0.658363\pi$
$284$	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	$-0.662444\pi$
$293$	−16.7224	−0.976934	−0.488467	+	0.872583i	$0.662444\pi$
$294$	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	$-0.414905\pi$
$307$	9.25690	0.528319	0.264159	+	0.964479i	$0.414905\pi$
$308$	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	$-0.482495\pi$
$317$	1.95728	0.109932	0.0549660	+	0.998488i	$0.482495\pi$
$318$	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	$-0.570212\pi$
$331$	−7.96122	−0.437588	−0.218794	+	0.975771i	$0.570212\pi$
$332$	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	$-0.465639\pi$
$347$	4.01387	0.215476	0.107738	+	0.994179i	$0.465639\pi$
$348$	0	0
$349$	−17.6744	−0.946091	−0.473045	−	0.881038i	$-0.656845\pi$
$349$	−17.6744	−0.946091	−0.473045	+	0.881038i	$0.656845\pi$
$350$	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	$-0.622644\pi$
$373$	−14.5171	−0.751669	−0.375834	+	0.926687i	$0.622644\pi$
$374$	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	$-0.619863\pi$
$379$	−14.3177	−0.735451	−0.367725	+	0.929934i	$0.619863\pi$
$380$	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	$-0.294561\pi$
$389$	23.7277	1.20304	0.601521	+	0.798857i	$0.294561\pi$
$390$	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	$-0.282403\pi$
$397$	25.1687	1.26318	0.631590	+	0.775303i	$0.282403\pi$
$398$	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	$-0.230911\pi$
$419$	30.6312	1.49643	0.748215	+	0.663456i	$0.230911\pi$
$420$	0	0
$421$	−16.8111	−0.819324	−0.409662	−	0.912237i	$-0.634353\pi$
$421$	−16.8111	−0.819324	−0.409662	+	0.912237i	$0.634353\pi$
$422$	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	$-0.562055\pi$
$443$	−8.15463	−0.387438	−0.193719	+	0.981057i	$0.562055\pi$
$444$	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	$-0.696654\pi$
$461$	−24.8740	−1.15850	−0.579248	+	0.815151i	$0.696654\pi$
$462$	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	$-0.757951\pi$
$467$	−31.3152	−1.44909	−0.724547	+	0.689225i	$0.757951\pi$
$468$	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	$-0.248477\pi$
$491$	31.4864	1.42096	0.710481	+	0.703716i	$0.248477\pi$
$492$	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	$-0.422305\pi$
$499$	10.7970	0.483339	0.241669	+	0.970359i	$0.422305\pi$
$500$	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	$-0.392037\pi$
$509$	15.0126	0.665421	0.332711	+	0.943029i	$0.392037\pi$
$510$	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	$-0.421345\pi$
$523$	11.1874	0.489189	0.244594	+	0.969625i	$0.421345\pi$
$524$	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	$-0.538203\pi$
$541$	−5.56974	−0.239462	−0.119731	+	0.992806i	$0.538203\pi$
$542$	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	$-0.170394\pi$
$547$	40.2326	1.72022	0.860111	+	0.510107i	$0.170394\pi$
$548$	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	$-0.514702\pi$
$557$	−2.17930	−0.0923398	−0.0461699	+	0.998934i	$0.514702\pi$
$558$	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	$-0.316353\pi$
$563$	25.8852	1.09093	0.545466	+	0.838133i	$0.316353\pi$
$564$	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	$-0.531135\pi$
$571$	−4.66716	−0.195314	−0.0976572	+	0.995220i	$0.531135\pi$
$572$	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	$-0.696563\pi$
$587$	−28.0569	−1.15803	−0.579015	+	0.815317i	$0.696563\pi$
$588$	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	$-0.295928\pi$
$613$	29.6158	1.19617	0.598086	+	0.801432i	$0.295928\pi$
$614$	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	$-0.289372\pi$
$619$	30.5755	1.22893	0.614466	+	0.788943i	$0.289372\pi$
$620$	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	$-0.488133\pi$
$643$	1.89024	0.0745436	0.0372718	+	0.999305i	$0.488133\pi$
$644$	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	$-0.555568\pi$
$653$	−8.87678	−0.347375	−0.173688	+	0.984801i	$0.555568\pi$
$654$	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	$-0.652702\pi$
$659$	−23.6963	−0.923077	−0.461538	+	0.887120i	$0.652702\pi$
$660$	0	0
$661$	−8.95134	−0.348167	−0.174083	−	0.984731i	$-0.555696\pi$
$661$	−8.95134	−0.348167	−0.174083	+	0.984731i	$0.555696\pi$
$662$	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	$-0.701828\pi$
$677$	−30.8288	−1.18485	−0.592423	+	0.805627i	$0.701828\pi$
$678$	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	$-0.406569\pi$
$683$	15.1226	0.578650	0.289325	+	0.957231i	$0.406569\pi$
$684$	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	$-0.525296\pi$
$691$	−4.17367	−0.158774	−0.0793870	+	0.996844i	$0.525296\pi$
$692$	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	$-0.513375\pi$
$701$	−2.22431	−0.0840110	−0.0420055	+	0.999117i	$0.513375\pi$
$702$	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	$-0.675672\pi$
$709$	−27.9210	−1.04859	−0.524297	+	0.851535i	$0.675672\pi$
$710$	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	$-0.640016\pi$
$733$	−23.0576	−0.851652	−0.425826	+	0.904805i	$0.640016\pi$
$734$	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	$-0.806081\pi$
$739$	−44.5880	−1.64019	−0.820097	+	0.572224i	$0.806081\pi$
$740$	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	$-0.329991\pi$
$757$	28.0125	1.01813	0.509066	+	0.860727i	$0.329991\pi$
$758$	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	$-0.404764\pi$
$773$	16.3897	0.589497	0.294749	+	0.955575i	$0.404764\pi$
$774$	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	$-0.519960\pi$
$787$	−3.51597	−0.125331	−0.0626654	+	0.998035i	$0.519960\pi$
$788$	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	$-0.616392\pi$
$797$	−20.1889	−0.715127	−0.357563	+	0.933889i	$0.616392\pi$
$798$	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	$-0.761047\pi$
$811$	−41.6472	−1.46243	−0.731216	+	0.682146i	$0.761047\pi$
$812$	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	$-0.628977\pi$
$821$	−22.5899	−0.788392	−0.394196	+	0.919026i	$0.628977\pi$
$822$	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	$-0.463399\pi$
$827$	6.59880	0.229463	0.114731	+	0.993397i	$0.463399\pi$
$828$	0	0
$829$	−11.6230	−0.403683	−0.201841	−	0.979418i	$-0.564693\pi$
$829$	−11.6230	−0.403683	−0.201841	+	0.979418i	$0.564693\pi$
$830$	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	$-0.934625\pi$
$853$	−57.1847	−1.95797	−0.978983	+	0.203941i	$0.934625\pi$
$854$	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	$-0.512439\pi$
$859$	−2.29004	−0.0781351	−0.0390676	+	0.999237i	$0.512439\pi$
$860$	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	$-0.774281\pi$
$877$	−44.9506	−1.51787	−0.758936	+	0.651165i	$0.774281\pi$
$878$	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	$-0.607052\pi$
$883$	−19.6126	−0.660018	−0.330009	+	0.943978i	$0.607052\pi$
$884$	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	$-0.0962538\pi$
$907$	57.5000	1.90925	0.954627	+	0.297803i	$0.0962538\pi$
$908$	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	$-0.532868\pi$
$941$	−6.32369	−0.206146	−0.103073	+	0.994674i	$0.532868\pi$
$942$	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	$-0.718308\pi$
$947$	−38.9787	−1.26664	−0.633319	+	0.773891i	$0.718308\pi$
$948$	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	$-0.626432\pi$
$971$	−24.1082	−0.773670	−0.386835	+	0.922149i	$0.626432\pi$
$972$	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	$-0.234986\pi$
$997$	46.7100	1.47932	0.739661	+	0.672980i	$0.234986\pi$
$998$	0	0
$999$	21.5926	0.683160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.2.a.h.1.1	yes	5
3.2	odd	2		8352.2.a.bh.1.3		5
4.3	odd	2		928.2.a.g.1.5	✓	5
8.3	odd	2		1856.2.a.bb.1.1		5
8.5	even	2		1856.2.a.ba.1.5		5
12.11	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	4.3	odd	2
928.2.a.h.1.1	yes	5	1.1	even	1	trivial
1856.2.a.ba.1.5		5	8.5	even	2
1856.2.a.bb.1.1		5	8.3	odd	2
8352.2.a.bg.1.3		5	12.11	even	2
8352.2.a.bh.1.3		5	3.2	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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	928.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

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