LMFDB - Embedded newform 4008.2.a.l.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.619387$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.619387 q^{5} +0.954126 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.619387 q^{5} +0.954126 q^{7} +1.00000 q^{9} +6.25107 q^{11} +2.92641 q^{13} -0.619387 q^{15} +5.33666 q^{17} +2.94423 q^{19} +0.954126 q^{21} -2.33637 q^{23} -4.61636 q^{25} +1.00000 q^{27} +4.47124 q^{29} -3.45595 q^{31} +6.25107 q^{33} -0.590974 q^{35} +4.26640 q^{37} +2.92641 q^{39} -10.8203 q^{41} -1.32390 q^{43} -0.619387 q^{45} +3.37095 q^{47} -6.08964 q^{49} +5.33666 q^{51} +5.72269 q^{53} -3.87184 q^{55} +2.94423 q^{57} -10.0347 q^{59} +11.8490 q^{61} +0.954126 q^{63} -1.81258 q^{65} +0.604492 q^{67} -2.33637 q^{69} -9.74739 q^{71} -11.1654 q^{73} -4.61636 q^{75} +5.96431 q^{77} +6.74411 q^{79} +1.00000 q^{81} +0.00272570 q^{83} -3.30546 q^{85} +4.47124 q^{87} +9.85457 q^{89} +2.79217 q^{91} -3.45595 q^{93} -1.82362 q^{95} -8.92479 q^{97} +6.25107 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.619387	−0.276999	−0.138499	−	0.990363i	$-0.544228\pi$
$5$	−0.619387	−0.276999	−0.138499	+	0.990363i	$0.544228\pi$
$6$	0	0
$7$	0.954126	0.360626	0.180313	−	0.983609i	$-0.442289\pi$
$7$	0.954126	0.360626	0.180313	+	0.983609i	$0.442289\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.25107	1.88477	0.942385	−	0.334531i	$-0.108578\pi$
$11$	6.25107	1.88477	0.942385	+	0.334531i	$0.108578\pi$
$12$	0	0
$13$	2.92641	0.811641	0.405820	−	0.913953i	$-0.366986\pi$
$13$	2.92641	0.811641	0.405820	+	0.913953i	$0.366986\pi$
$14$	0	0
$15$	−0.619387	−0.159925
$16$	0	0
$17$	5.33666	1.29433	0.647165	−	0.762350i	$-0.275954\pi$
$17$	5.33666	1.29433	0.647165	+	0.762350i	$0.275954\pi$
$18$	0	0
$19$	2.94423	0.675454	0.337727	−	0.941244i	$-0.390342\pi$
$19$	2.94423	0.675454	0.337727	+	0.941244i	$0.390342\pi$
$20$	0	0
$21$	0.954126	0.208207
$22$	0	0
$23$	−2.33637	−0.487167	−0.243583	−	0.969880i	$-0.578323\pi$
$23$	−2.33637	−0.487167	−0.243583	+	0.969880i	$0.578323\pi$
$24$	0	0
$25$	−4.61636	−0.923272
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.47124	0.830289	0.415144	−	0.909756i	$-0.363731\pi$
$29$	4.47124	0.830289	0.415144	+	0.909756i	$0.363731\pi$
$30$	0	0
$31$	−3.45595	−0.620707	−0.310353	−	0.950621i	$-0.600447\pi$
$31$	−3.45595	−0.620707	−0.310353	+	0.950621i	$0.600447\pi$
$32$	0	0
$33$	6.25107	1.08817
$34$	0	0
$35$	−0.590974	−0.0998928
$36$	0	0
$37$	4.26640	0.701392	0.350696	−	0.936489i	$-0.385945\pi$
$37$	4.26640	0.701392	0.350696	+	0.936489i	$0.385945\pi$
$38$	0	0
$39$	2.92641	0.468601
$40$	0	0
$41$	−10.8203	−1.68985	−0.844925	−	0.534885i	$-0.820355\pi$
$41$	−10.8203	−1.68985	−0.844925	+	0.534885i	$0.820355\pi$
$42$	0	0
$43$	−1.32390	−0.201893	−0.100947	−	0.994892i	$-0.532187\pi$
$43$	−1.32390	−0.201893	−0.100947	+	0.994892i	$0.532187\pi$
$44$	0	0
$45$	−0.619387	−0.0923328
$46$	0	0
$47$	3.37095	0.491704	0.245852	−	0.969307i	$-0.420932\pi$
$47$	3.37095	0.491704	0.245852	+	0.969307i	$0.420932\pi$
$48$	0	0
$49$	−6.08964	−0.869949
$50$	0	0
$51$	5.33666	0.747282
$52$	0	0
$53$	5.72269	0.786072	0.393036	−	0.919523i	$-0.371425\pi$
$53$	5.72269	0.786072	0.393036	+	0.919523i	$0.371425\pi$
$54$	0	0
$55$	−3.87184	−0.522078
$56$	0	0
$57$	2.94423	0.389973
$58$	0	0
$59$	−10.0347	−1.30641	−0.653204	−	0.757182i	$-0.726575\pi$
$59$	−10.0347	−1.30641	−0.653204	+	0.757182i	$0.726575\pi$
$60$	0	0
$61$	11.8490	1.51711	0.758556	−	0.651608i	$-0.225905\pi$
$61$	11.8490	1.51711	0.758556	+	0.651608i	$0.225905\pi$
$62$	0	0
$63$	0.954126	0.120209
$64$	0	0
$65$	−1.81258	−0.224823
$66$	0	0
$67$	0.604492	0.0738505	0.0369252	−	0.999318i	$-0.488244\pi$
$67$	0.604492	0.0738505	0.0369252	+	0.999318i	$0.488244\pi$
$68$	0	0
$69$	−2.33637	−0.281266
$70$	0	0
$71$	−9.74739	−1.15680	−0.578401	−	0.815752i	$-0.696323\pi$
$71$	−9.74739	−1.15680	−0.578401	+	0.815752i	$0.696323\pi$
$72$	0	0
$73$	−11.1654	−1.30681	−0.653404	−	0.757009i	$-0.726660\pi$
$73$	−11.1654	−1.30681	−0.653404	+	0.757009i	$0.726660\pi$
$74$	0	0
$75$	−4.61636	−0.533051
$76$	0	0
$77$	5.96431	0.679697
$78$	0	0
$79$	6.74411	0.758772	0.379386	−	0.925239i	$-0.376135\pi$
$79$	6.74411	0.758772	0.379386	+	0.925239i	$0.376135\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.00272570	0.000299184 0	0.000149592	−	1.00000i	$-0.499952\pi$
$83$	0.00272570	0.000299184 0	0.000149592		1.00000i	$0.499952\pi$
$84$	0	0
$85$	−3.30546	−0.358528
$86$	0	0
$87$	4.47124	0.479367
$88$	0	0
$89$	9.85457	1.04458	0.522291	−	0.852767i	$-0.325077\pi$
$89$	9.85457	1.04458	0.522291	+	0.852767i	$0.325077\pi$
$90$	0	0
$91$	2.79217	0.292699
$92$	0	0
$93$	−3.45595	−0.358365
$94$	0	0
$95$	−1.82362	−0.187100
$96$	0	0
$97$	−8.92479	−0.906175	−0.453088	−	0.891466i	$-0.649678\pi$
$97$	−8.92479	−0.906175	−0.453088	+	0.891466i	$0.649678\pi$
$98$	0	0
$99$	6.25107	0.628256
$100$	0	0
$101$	1.91531	0.190580	0.0952901	−	0.995450i	$-0.469622\pi$
$101$	1.91531	0.190580	0.0952901	+	0.995450i	$0.469622\pi$
$102$	0	0
$103$	5.90694	0.582028	0.291014	−	0.956719i	$-0.406007\pi$
$103$	5.90694	0.582028	0.291014	+	0.956719i	$0.406007\pi$
$104$	0	0
$105$	−0.590974	−0.0576732
$106$	0	0
$107$	−7.36790	−0.712282	−0.356141	−	0.934432i	$-0.615908\pi$
$107$	−7.36790	−0.712282	−0.356141	+	0.934432i	$0.615908\pi$
$108$	0	0
$109$	9.19031	0.880272	0.440136	−	0.897931i	$-0.354930\pi$
$109$	9.19031	0.880272	0.440136	+	0.897931i	$0.354930\pi$
$110$	0	0
$111$	4.26640	0.404949
$112$	0	0
$113$	−12.9146	−1.21490	−0.607451	−	0.794357i	$-0.707808\pi$
$113$	−12.9146	−1.21490	−0.607451	+	0.794357i	$0.707808\pi$
$114$	0	0
$115$	1.44712	0.134944
$116$	0	0
$117$	2.92641	0.270547
$118$	0	0
$119$	5.09185	0.466769
$120$	0	0
$121$	28.0759	2.55236
$122$	0	0
$123$	−10.8203	−0.975635
$124$	0	0
$125$	5.95625	0.532743
$126$	0	0
$127$	9.28542	0.823948	0.411974	−	0.911196i	$-0.364839\pi$
$127$	9.28542	0.823948	0.411974	+	0.911196i	$0.364839\pi$
$128$	0	0
$129$	−1.32390	−0.116563
$130$	0	0
$131$	−2.24628	−0.196259	−0.0981293	−	0.995174i	$-0.531286\pi$
$131$	−2.24628	−0.196259	−0.0981293	+	0.995174i	$0.531286\pi$
$132$	0	0
$133$	2.80917	0.243586
$134$	0	0
$135$	−0.619387	−0.0533084
$136$	0	0
$137$	−0.893884	−0.0763696	−0.0381848	−	0.999271i	$-0.512158\pi$
$137$	−0.893884	−0.0763696	−0.0381848	+	0.999271i	$0.512158\pi$
$138$	0	0
$139$	1.11800	0.0948276	0.0474138	−	0.998875i	$-0.484902\pi$
$139$	1.11800	0.0948276	0.0474138	+	0.998875i	$0.484902\pi$
$140$	0	0
$141$	3.37095	0.283885
$142$	0	0
$143$	18.2932	1.52976
$144$	0	0
$145$	−2.76943	−0.229989
$146$	0	0
$147$	−6.08964	−0.502265
$148$	0	0
$149$	−10.8738	−0.890814	−0.445407	−	0.895328i	$-0.646941\pi$
$149$	−10.8738	−0.890814	−0.445407	+	0.895328i	$0.646941\pi$
$150$	0	0
$151$	23.9232	1.94684	0.973421	−	0.229023i	$-0.0735530\pi$
$151$	23.9232	1.94684	0.973421	+	0.229023i	$0.0735530\pi$
$152$	0	0
$153$	5.33666	0.431443
$154$	0	0
$155$	2.14057	0.171935
$156$	0	0
$157$	10.4595	0.834758	0.417379	−	0.908733i	$-0.362949\pi$
$157$	10.4595	0.834758	0.417379	+	0.908733i	$0.362949\pi$
$158$	0	0
$159$	5.72269	0.453839
$160$	0	0
$161$	−2.22919	−0.175685
$162$	0	0
$163$	−1.79788	−0.140821	−0.0704103	−	0.997518i	$-0.522431\pi$
$163$	−1.79788	−0.140821	−0.0704103	+	0.997518i	$0.522431\pi$
$164$	0	0
$165$	−3.87184	−0.301422
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−4.43611	−0.341240
$170$	0	0
$171$	2.94423	0.225151
$172$	0	0
$173$	6.35455	0.483127	0.241564	−	0.970385i	$-0.422340\pi$
$173$	6.35455	0.483127	0.241564	+	0.970385i	$0.422340\pi$
$174$	0	0
$175$	−4.40459	−0.332956
$176$	0	0
$177$	−10.0347	−0.754255
$178$	0	0
$179$	15.3384	1.14645	0.573224	−	0.819399i	$-0.305693\pi$
$179$	15.3384	1.14645	0.573224	+	0.819399i	$0.305693\pi$
$180$	0	0
$181$	2.95349	0.219531	0.109766	−	0.993958i	$-0.464990\pi$
$181$	2.95349	0.219531	0.109766	+	0.993958i	$0.464990\pi$
$182$	0	0
$183$	11.8490	0.875904
$184$	0	0
$185$	−2.64256	−0.194285
$186$	0	0
$187$	33.3598	2.43951
$188$	0	0
$189$	0.954126	0.0694025
$190$	0	0
$191$	1.76766	0.127903	0.0639515	−	0.997953i	$-0.479630\pi$
$191$	1.76766	0.127903	0.0639515	+	0.997953i	$0.479630\pi$
$192$	0	0
$193$	−11.2653	−0.810897	−0.405448	−	0.914118i	$-0.632885\pi$
$193$	−11.2653	−0.810897	−0.405448	+	0.914118i	$0.632885\pi$
$194$	0	0
$195$	−1.81258	−0.129802
$196$	0	0
$197$	−20.9384	−1.49180	−0.745901	−	0.666057i	$-0.767981\pi$
$197$	−20.9384	−1.49180	−0.745901	+	0.666057i	$0.767981\pi$
$198$	0	0
$199$	12.8279	0.909347	0.454674	−	0.890658i	$-0.349756\pi$
$199$	12.8279	0.909347	0.454674	+	0.890658i	$0.349756\pi$
$200$	0	0
$201$	0.604492	0.0426376
$202$	0	0
$203$	4.26613	0.299424
$204$	0	0
$205$	6.70197	0.468086
$206$	0	0
$207$	−2.33637	−0.162389
$208$	0	0
$209$	18.4046	1.27307
$210$	0	0
$211$	−8.08886	−0.556860	−0.278430	−	0.960457i	$-0.589814\pi$
$211$	−8.08886	−0.556860	−0.278430	+	0.960457i	$0.589814\pi$
$212$	0	0
$213$	−9.74739	−0.667880
$214$	0	0
$215$	0.820009	0.0559242
$216$	0	0
$217$	−3.29741	−0.223843
$218$	0	0
$219$	−11.1654	−0.754486
$220$	0	0
$221$	15.6173	1.05053
$222$	0	0
$223$	24.7517	1.65750	0.828749	−	0.559620i	$-0.189053\pi$
$223$	24.7517	1.65750	0.828749	+	0.559620i	$0.189053\pi$
$224$	0	0
$225$	−4.61636	−0.307757
$226$	0	0
$227$	−1.26183	−0.0837504	−0.0418752	−	0.999123i	$-0.513333\pi$
$227$	−1.26183	−0.0837504	−0.0418752	+	0.999123i	$0.513333\pi$
$228$	0	0
$229$	12.5768	0.831096	0.415548	−	0.909571i	$-0.363590\pi$
$229$	12.5768	0.831096	0.415548	+	0.909571i	$0.363590\pi$
$230$	0	0
$231$	5.96431	0.392423
$232$	0	0
$233$	15.6215	1.02340	0.511698	−	0.859165i	$-0.329017\pi$
$233$	15.6215	1.02340	0.511698	+	0.859165i	$0.329017\pi$
$234$	0	0
$235$	−2.08793	−0.136201
$236$	0	0
$237$	6.74411	0.438077
$238$	0	0
$239$	−27.8379	−1.80068	−0.900341	−	0.435185i	$-0.856683\pi$
$239$	−27.8379	−1.80068	−0.900341	+	0.435185i	$0.856683\pi$
$240$	0	0
$241$	−31.0024	−1.99704	−0.998521	−	0.0543633i	$-0.982687\pi$
$241$	−31.0024	−1.99704	−0.998521	+	0.0543633i	$0.982687\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.77185	0.240975
$246$	0	0
$247$	8.61604	0.548226
$248$	0	0
$249$	0.00272570	0.000172734 0
$250$	0	0
$251$	2.88623	0.182177	0.0910885	−	0.995843i	$-0.470965\pi$
$251$	2.88623	0.182177	0.0910885	+	0.995843i	$0.470965\pi$
$252$	0	0
$253$	−14.6048	−0.918197
$254$	0	0
$255$	−3.30546	−0.206996
$256$	0	0
$257$	−10.5265	−0.656626	−0.328313	−	0.944569i	$-0.606480\pi$
$257$	−10.5265	−0.656626	−0.328313	+	0.944569i	$0.606480\pi$
$258$	0	0
$259$	4.07069	0.252940
$260$	0	0
$261$	4.47124	0.276763
$262$	0	0
$263$	−13.9049	−0.857415	−0.428708	−	0.903443i	$-0.641031\pi$
$263$	−13.9049	−0.857415	−0.428708	+	0.903443i	$0.641031\pi$
$264$	0	0
$265$	−3.54456	−0.217741
$266$	0	0
$267$	9.85457	0.603090
$268$	0	0
$269$	8.54423	0.520951	0.260475	−	0.965480i	$-0.416121\pi$
$269$	8.54423	0.520951	0.260475	+	0.965480i	$0.416121\pi$
$270$	0	0
$271$	3.35225	0.203635	0.101817	−	0.994803i	$-0.467534\pi$
$271$	3.35225	0.203635	0.101817	+	0.994803i	$0.467534\pi$
$272$	0	0
$273$	2.79217	0.168990
$274$	0	0
$275$	−28.8572	−1.74015
$276$	0	0
$277$	−11.5788	−0.695700	−0.347850	−	0.937550i	$-0.613088\pi$
$277$	−11.5788	−0.695700	−0.347850	+	0.937550i	$0.613088\pi$
$278$	0	0
$279$	−3.45595	−0.206902
$280$	0	0
$281$	−7.10103	−0.423612	−0.211806	−	0.977312i	$-0.567935\pi$
$281$	−7.10103	−0.423612	−0.211806	+	0.977312i	$0.567935\pi$
$282$	0	0
$283$	−3.58868	−0.213325	−0.106662	−	0.994295i	$-0.534016\pi$
$283$	−3.58868	−0.213325	−0.106662	+	0.994295i	$0.534016\pi$
$284$	0	0
$285$	−1.82362	−0.108022
$286$	0	0
$287$	−10.3240	−0.609404
$288$	0	0
$289$	11.4799	0.675291
$290$	0	0
$291$	−8.92479	−0.523180
$292$	0	0
$293$	−23.4842	−1.37196	−0.685982	−	0.727619i	$-0.740627\pi$
$293$	−23.4842	−1.37196	−0.685982	+	0.727619i	$0.740627\pi$
$294$	0	0
$295$	6.21537	0.361873
$296$	0	0
$297$	6.25107	0.362724
$298$	0	0
$299$	−6.83718	−0.395404
$300$	0	0
$301$	−1.26317	−0.0728080
$302$	0	0
$303$	1.91531	0.110031
$304$	0	0
$305$	−7.33913	−0.420237
$306$	0	0
$307$	3.96744	0.226433	0.113217	−	0.993570i	$-0.463885\pi$
$307$	3.96744	0.226433	0.113217	+	0.993570i	$0.463885\pi$
$308$	0	0
$309$	5.90694	0.336034
$310$	0	0
$311$	1.69973	0.0963829	0.0481915	−	0.998838i	$-0.484654\pi$
$311$	1.69973	0.0963829	0.0481915	+	0.998838i	$0.484654\pi$
$312$	0	0
$313$	9.65206	0.545566	0.272783	−	0.962076i	$-0.412056\pi$
$313$	9.65206	0.545566	0.272783	+	0.962076i	$0.412056\pi$
$314$	0	0
$315$	−0.590974	−0.0332976
$316$	0	0
$317$	−8.58465	−0.482162	−0.241081	−	0.970505i	$-0.577502\pi$
$317$	−8.58465	−0.482162	−0.241081	+	0.970505i	$0.577502\pi$
$318$	0	0
$319$	27.9501	1.56490
$320$	0	0
$321$	−7.36790	−0.411236
$322$	0	0
$323$	15.7124	0.874260
$324$	0	0
$325$	−13.5094	−0.749365
$326$	0	0
$327$	9.19031	0.508225
$328$	0	0
$329$	3.21631	0.177321
$330$	0	0
$331$	13.0207	0.715683	0.357841	−	0.933782i	$-0.383513\pi$
$331$	13.0207	0.715683	0.357841	+	0.933782i	$0.383513\pi$
$332$	0	0
$333$	4.26640	0.233797
$334$	0	0
$335$	−0.374415	−0.0204565
$336$	0	0
$337$	1.58095	0.0861199	0.0430600	−	0.999072i	$-0.486289\pi$
$337$	1.58095	0.0861199	0.0430600	+	0.999072i	$0.486289\pi$
$338$	0	0
$339$	−12.9146	−0.701424
$340$	0	0
$341$	−21.6034	−1.16989
$342$	0	0
$343$	−12.4892	−0.674352
$344$	0	0
$345$	1.44712	0.0779102
$346$	0	0
$347$	20.4617	1.09844	0.549220	−	0.835678i	$-0.314925\pi$
$347$	20.4617	1.09844	0.549220	+	0.835678i	$0.314925\pi$
$348$	0	0
$349$	3.71699	0.198966	0.0994831	−	0.995039i	$-0.468281\pi$
$349$	3.71699	0.198966	0.0994831	+	0.995039i	$0.468281\pi$
$350$	0	0
$351$	2.92641	0.156200
$352$	0	0
$353$	22.2105	1.18214	0.591072	−	0.806619i	$-0.298705\pi$
$353$	22.2105	1.18214	0.591072	+	0.806619i	$0.298705\pi$
$354$	0	0
$355$	6.03741	0.320433
$356$	0	0
$357$	5.09185	0.269489
$358$	0	0
$359$	−3.53420	−0.186528	−0.0932640	−	0.995641i	$-0.529730\pi$
$359$	−3.53420	−0.186528	−0.0932640	+	0.995641i	$0.529730\pi$
$360$	0	0
$361$	−10.3315	−0.543762
$362$	0	0
$363$	28.0759	1.47360
$364$	0	0
$365$	6.91569	0.361984
$366$	0	0
$367$	10.2786	0.536537	0.268268	−	0.963344i	$-0.413549\pi$
$367$	10.2786	0.536537	0.268268	+	0.963344i	$0.413549\pi$
$368$	0	0
$369$	−10.8203	−0.563283
$370$	0	0
$371$	5.46017	0.283478
$372$	0	0
$373$	−22.5828	−1.16929	−0.584647	−	0.811288i	$-0.698767\pi$
$373$	−22.5828	−1.16929	−0.584647	+	0.811288i	$0.698767\pi$
$374$	0	0
$375$	5.95625	0.307580
$376$	0	0
$377$	13.0847	0.673896
$378$	0	0
$379$	10.3926	0.533834	0.266917	−	0.963720i	$-0.413995\pi$
$379$	10.3926	0.533834	0.266917	+	0.963720i	$0.413995\pi$
$380$	0	0
$381$	9.28542	0.475707
$382$	0	0
$383$	6.01014	0.307104	0.153552	−	0.988141i	$-0.450929\pi$
$383$	6.01014	0.307104	0.153552	+	0.988141i	$0.450929\pi$
$384$	0	0
$385$	−3.69422	−0.188275
$386$	0	0
$387$	−1.32390	−0.0672978
$388$	0	0
$389$	−22.9823	−1.16525	−0.582626	−	0.812741i	$-0.697975\pi$
$389$	−22.9823	−1.16525	−0.582626	+	0.812741i	$0.697975\pi$
$390$	0	0
$391$	−12.4684	−0.630555
$392$	0	0
$393$	−2.24628	−0.113310
$394$	0	0
$395$	−4.17722	−0.210179
$396$	0	0
$397$	−38.2310	−1.91876	−0.959381	−	0.282115i	$-0.908964\pi$
$397$	−38.2310	−1.91876	−0.959381	+	0.282115i	$0.908964\pi$
$398$	0	0
$399$	2.80917	0.140634
$400$	0	0
$401$	−22.9593	−1.14653	−0.573266	−	0.819370i	$-0.694324\pi$
$401$	−22.9593	−1.14653	−0.573266	+	0.819370i	$0.694324\pi$
$402$	0	0
$403$	−10.1135	−0.503791
$404$	0	0
$405$	−0.619387	−0.0307776
$406$	0	0
$407$	26.6696	1.32196
$408$	0	0
$409$	21.3129	1.05386	0.526928	−	0.849910i	$-0.323344\pi$
$409$	21.3129	1.05386	0.526928	+	0.849910i	$0.323344\pi$
$410$	0	0
$411$	−0.893884	−0.0440920
$412$	0	0
$413$	−9.57438	−0.471124
$414$	0	0
$415$	−0.00168826	−8.28736e−5 0
$416$	0	0
$417$	1.11800	0.0547487
$418$	0	0
$419$	−31.8211	−1.55456	−0.777281	−	0.629154i	$-0.783402\pi$
$419$	−31.8211	−1.55456	−0.777281	+	0.629154i	$0.783402\pi$
$420$	0	0
$421$	−0.234009	−0.0114049	−0.00570245	−	0.999984i	$-0.501815\pi$
$421$	−0.234009	−0.0114049	−0.00570245	+	0.999984i	$0.501815\pi$
$422$	0	0
$423$	3.37095	0.163901
$424$	0	0
$425$	−24.6359	−1.19502
$426$	0	0
$427$	11.3055	0.547110
$428$	0	0
$429$	18.2932	0.883205
$430$	0	0
$431$	−9.97830	−0.480638	−0.240319	−	0.970694i	$-0.577252\pi$
$431$	−9.97830	−0.480638	−0.240319	+	0.970694i	$0.577252\pi$
$432$	0	0
$433$	31.2262	1.50064	0.750318	−	0.661077i	$-0.229901\pi$
$433$	31.2262	1.50064	0.750318	+	0.661077i	$0.229901\pi$
$434$	0	0
$435$	−2.76943	−0.132784
$436$	0	0
$437$	−6.87882	−0.329059
$438$	0	0
$439$	−15.4486	−0.737320	−0.368660	−	0.929564i	$-0.620183\pi$
$439$	−15.4486	−0.737320	−0.368660	+	0.929564i	$0.620183\pi$
$440$	0	0
$441$	−6.08964	−0.289983
$442$	0	0
$443$	−4.04906	−0.192377	−0.0961883	−	0.995363i	$-0.530665\pi$
$443$	−4.04906	−0.192377	−0.0961883	+	0.995363i	$0.530665\pi$
$444$	0	0
$445$	−6.10380	−0.289348
$446$	0	0
$447$	−10.8738	−0.514312
$448$	0	0
$449$	−3.16361	−0.149300	−0.0746500	−	0.997210i	$-0.523784\pi$
$449$	−3.16361	−0.149300	−0.0746500	+	0.997210i	$0.523784\pi$
$450$	0	0
$451$	−67.6386	−3.18498
$452$	0	0
$453$	23.9232	1.12401
$454$	0	0
$455$	−1.72943	−0.0810771
$456$	0	0
$457$	−21.9553	−1.02702	−0.513512	−	0.858083i	$-0.671656\pi$
$457$	−21.9553	−1.02702	−0.513512	+	0.858083i	$0.671656\pi$
$458$	0	0
$459$	5.33666	0.249094
$460$	0	0
$461$	−25.4993	−1.18762	−0.593811	−	0.804604i	$-0.702378\pi$
$461$	−25.4993	−1.18762	−0.593811	+	0.804604i	$0.702378\pi$
$462$	0	0
$463$	0.891709	0.0414412	0.0207206	−	0.999785i	$-0.493404\pi$
$463$	0.891709	0.0414412	0.0207206	+	0.999785i	$0.493404\pi$
$464$	0	0
$465$	2.14057	0.0992666
$466$	0	0
$467$	30.2712	1.40079	0.700393	−	0.713758i	$-0.253008\pi$
$467$	30.2712	1.40079	0.700393	+	0.713758i	$0.253008\pi$
$468$	0	0
$469$	0.576762	0.0266324
$470$	0	0
$471$	10.4595	0.481948
$472$	0	0
$473$	−8.27582	−0.380523
$474$	0	0
$475$	−13.5916	−0.623627
$476$	0	0
$477$	5.72269	0.262024
$478$	0	0
$479$	−1.73323	−0.0791932	−0.0395966	−	0.999216i	$-0.512607\pi$
$479$	−1.73323	−0.0791932	−0.0395966	+	0.999216i	$0.512607\pi$
$480$	0	0
$481$	12.4852	0.569278
$482$	0	0
$483$	−2.22919	−0.101432
$484$	0	0
$485$	5.52790	0.251009
$486$	0	0
$487$	41.7612	1.89238	0.946190	−	0.323612i	$-0.104897\pi$
$487$	41.7612	1.89238	0.946190	+	0.323612i	$0.104897\pi$
$488$	0	0
$489$	−1.79788	−0.0813029
$490$	0	0
$491$	34.6055	1.56173	0.780863	−	0.624703i	$-0.214780\pi$
$491$	34.6055	1.56173	0.780863	+	0.624703i	$0.214780\pi$
$492$	0	0
$493$	23.8615	1.07467
$494$	0	0
$495$	−3.87184	−0.174026
$496$	0	0
$497$	−9.30024	−0.417173
$498$	0	0
$499$	17.8402	0.798636	0.399318	−	0.916813i	$-0.369247\pi$
$499$	17.8402	0.798636	0.399318	+	0.916813i	$0.369247\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−25.0372	−1.11635	−0.558176	−	0.829722i	$-0.688499\pi$
$503$	−25.0372	−1.11635	−0.558176	+	0.829722i	$0.688499\pi$
$504$	0	0
$505$	−1.18632	−0.0527904
$506$	0	0
$507$	−4.43611	−0.197015
$508$	0	0
$509$	−16.2180	−0.718852	−0.359426	−	0.933174i	$-0.617027\pi$
$509$	−16.2180	−0.718852	−0.359426	+	0.933174i	$0.617027\pi$
$510$	0	0
$511$	−10.6532	−0.471269
$512$	0	0
$513$	2.94423	0.129991
$514$	0	0
$515$	−3.65868	−0.161221
$516$	0	0
$517$	21.0721	0.926748
$518$	0	0
$519$	6.35455	0.278934
$520$	0	0
$521$	−23.8985	−1.04701	−0.523506	−	0.852022i	$-0.675376\pi$
$521$	−23.8985	−1.04701	−0.523506	+	0.852022i	$0.675376\pi$
$522$	0	0
$523$	−7.56408	−0.330754	−0.165377	−	0.986230i	$-0.552884\pi$
$523$	−7.56408	−0.330754	−0.165377	+	0.986230i	$0.552884\pi$
$524$	0	0
$525$	−4.40459	−0.192232
$526$	0	0
$527$	−18.4432	−0.803399
$528$	0	0
$529$	−17.5414	−0.762669
$530$	0	0
$531$	−10.0347	−0.435469
$532$	0	0
$533$	−31.6647	−1.37155
$534$	0	0
$535$	4.56358	0.197301
$536$	0	0
$537$	15.3384	0.661901
$538$	0	0
$539$	−38.0668	−1.63965
$540$	0	0
$541$	5.93874	0.255326	0.127663	−	0.991818i	$-0.459252\pi$
$541$	5.93874	0.255326	0.127663	+	0.991818i	$0.459252\pi$
$542$	0	0
$543$	2.95349	0.126746
$544$	0	0
$545$	−5.69236	−0.243834
$546$	0	0
$547$	−14.2274	−0.608319	−0.304159	−	0.952621i	$-0.598376\pi$
$547$	−14.2274	−0.608319	−0.304159	+	0.952621i	$0.598376\pi$
$548$	0	0
$549$	11.8490	0.505704
$550$	0	0
$551$	13.1644	0.560822
$552$	0	0
$553$	6.43473	0.273633
$554$	0	0
$555$	−2.64256	−0.112170
$556$	0	0
$557$	43.7728	1.85471	0.927356	−	0.374179i	$-0.122076\pi$
$557$	43.7728	1.85471	0.927356	+	0.374179i	$0.122076\pi$
$558$	0	0
$559$	−3.87429	−0.163865
$560$	0	0
$561$	33.3598	1.40845
$562$	0	0
$563$	1.51748	0.0639541	0.0319770	−	0.999489i	$-0.489820\pi$
$563$	1.51748	0.0639541	0.0319770	+	0.999489i	$0.489820\pi$
$564$	0	0
$565$	7.99913	0.336526
$566$	0	0
$567$	0.954126	0.0400695
$568$	0	0
$569$	31.3447	1.31404	0.657019	−	0.753874i	$-0.271817\pi$
$569$	31.3447	1.31404	0.657019	+	0.753874i	$0.271817\pi$
$570$	0	0
$571$	−32.7936	−1.37237	−0.686184	−	0.727428i	$-0.740716\pi$
$571$	−32.7936	−1.37237	−0.686184	+	0.727428i	$0.740716\pi$
$572$	0	0
$573$	1.76766	0.0738449
$574$	0	0
$575$	10.7855	0.449787
$576$	0	0
$577$	17.2365	0.717565	0.358782	−	0.933421i	$-0.383192\pi$
$577$	17.2365	0.717565	0.358782	+	0.933421i	$0.383192\pi$
$578$	0	0
$579$	−11.2653	−0.468171
$580$	0	0
$581$	0.00260066	0.000107894 0
$582$	0	0
$583$	35.7729	1.48156
$584$	0	0
$585$	−1.81258	−0.0749411
$586$	0	0
$587$	14.9256	0.616045	0.308023	−	0.951379i	$-0.400333\pi$
$587$	14.9256	0.616045	0.308023	+	0.951379i	$0.400333\pi$
$588$	0	0
$589$	−10.1751	−0.419259
$590$	0	0
$591$	−20.9384	−0.861293
$592$	0	0
$593$	−1.00970	−0.0414635	−0.0207318	−	0.999785i	$-0.506600\pi$
$593$	−1.00970	−0.0414635	−0.0207318	+	0.999785i	$0.506600\pi$
$594$	0	0
$595$	−3.15383	−0.129294
$596$	0	0
$597$	12.8279	0.525012
$598$	0	0
$599$	1.66824	0.0681624	0.0340812	−	0.999419i	$-0.489150\pi$
$599$	1.66824	0.0681624	0.0340812	+	0.999419i	$0.489150\pi$
$600$	0	0
$601$	−21.4063	−0.873182	−0.436591	−	0.899660i	$-0.643814\pi$
$601$	−21.4063	−0.873182	−0.436591	+	0.899660i	$0.643814\pi$
$602$	0	0
$603$	0.604492	0.0246168
$604$	0	0
$605$	−17.3899	−0.706999
$606$	0	0
$607$	36.1471	1.46716	0.733582	−	0.679601i	$-0.237847\pi$
$607$	36.1471	1.46716	0.733582	+	0.679601i	$0.237847\pi$
$608$	0	0
$609$	4.26613	0.172872
$610$	0	0
$611$	9.86479	0.399087
$612$	0	0
$613$	−27.3272	−1.10373	−0.551867	−	0.833932i	$-0.686084\pi$
$613$	−27.3272	−1.10373	−0.551867	+	0.833932i	$0.686084\pi$
$614$	0	0
$615$	6.70197	0.270250
$616$	0	0
$617$	15.4624	0.622491	0.311246	−	0.950330i	$-0.399254\pi$
$617$	15.4624	0.622491	0.311246	+	0.950330i	$0.399254\pi$
$618$	0	0
$619$	9.42786	0.378938	0.189469	−	0.981887i	$-0.439323\pi$
$619$	9.42786	0.378938	0.189469	+	0.981887i	$0.439323\pi$
$620$	0	0
$621$	−2.33637	−0.0937553
$622$	0	0
$623$	9.40251	0.376704
$624$	0	0
$625$	19.3926	0.775703
$626$	0	0
$627$	18.4046	0.735010
$628$	0	0
$629$	22.7683	0.907833
$630$	0	0
$631$	−3.25940	−0.129755	−0.0648774	−	0.997893i	$-0.520666\pi$
$631$	−3.25940	−0.129755	−0.0648774	+	0.997893i	$0.520666\pi$
$632$	0	0
$633$	−8.08886	−0.321503
$634$	0	0
$635$	−5.75127	−0.228232
$636$	0	0
$637$	−17.8208	−0.706086
$638$	0	0
$639$	−9.74739	−0.385601
$640$	0	0
$641$	−32.6986	−1.29152	−0.645758	−	0.763542i	$-0.723458\pi$
$641$	−32.6986	−1.29152	−0.645758	+	0.763542i	$0.723458\pi$
$642$	0	0
$643$	43.9513	1.73327	0.866635	−	0.498943i	$-0.166278\pi$
$643$	43.9513	1.73327	0.866635	+	0.498943i	$0.166278\pi$
$644$	0	0
$645$	0.820009	0.0322878
$646$	0	0
$647$	−37.4757	−1.47332	−0.736661	−	0.676262i	$-0.763599\pi$
$647$	−37.4757	−1.47332	−0.736661	+	0.676262i	$0.763599\pi$
$648$	0	0
$649$	−62.7277	−2.46228
$650$	0	0
$651$	−3.29741	−0.129236
$652$	0	0
$653$	−32.0885	−1.25572	−0.627860	−	0.778326i	$-0.716069\pi$
$653$	−32.0885	−1.25572	−0.627860	+	0.778326i	$0.716069\pi$
$654$	0	0
$655$	1.39132	0.0543633
$656$	0	0
$657$	−11.1654	−0.435603
$658$	0	0
$659$	48.8689	1.90366	0.951831	−	0.306623i	$-0.0991990\pi$
$659$	48.8689	1.90366	0.951831	+	0.306623i	$0.0991990\pi$
$660$	0	0
$661$	−1.28040	−0.0498018	−0.0249009	−	0.999690i	$-0.507927\pi$
$661$	−1.28040	−0.0498018	−0.0249009	+	0.999690i	$0.507927\pi$
$662$	0	0
$663$	15.6173	0.606524
$664$	0	0
$665$	−1.73997	−0.0674730
$666$	0	0
$667$	−10.4465	−0.404489
$668$	0	0
$669$	24.7517	0.956957
$670$	0	0
$671$	74.0691	2.85940
$672$	0	0
$673$	7.53994	0.290643	0.145322	−	0.989384i	$-0.453578\pi$
$673$	7.53994	0.290643	0.145322	+	0.989384i	$0.453578\pi$
$674$	0	0
$675$	−4.61636	−0.177684
$676$	0	0
$677$	−29.9878	−1.15253	−0.576263	−	0.817264i	$-0.695490\pi$
$677$	−29.9878	−1.15253	−0.576263	+	0.817264i	$0.695490\pi$
$678$	0	0
$679$	−8.51538	−0.326790
$680$	0	0
$681$	−1.26183	−0.0483533
$682$	0	0
$683$	18.5606	0.710201	0.355100	−	0.934828i	$-0.384447\pi$
$683$	18.5606	0.710201	0.355100	+	0.934828i	$0.384447\pi$
$684$	0	0
$685$	0.553660	0.0211543
$686$	0	0
$687$	12.5768	0.479834
$688$	0	0
$689$	16.7469	0.638008
$690$	0	0
$691$	−25.4157	−0.966857	−0.483429	−	0.875384i	$-0.660609\pi$
$691$	−25.4157	−0.966857	−0.483429	+	0.875384i	$0.660609\pi$
$692$	0	0
$693$	5.96431	0.226566
$694$	0	0
$695$	−0.692476	−0.0262671
$696$	0	0
$697$	−57.7444	−2.18722
$698$	0	0
$699$	15.6215	0.590858
$700$	0	0
$701$	−1.05383	−0.0398026	−0.0199013	−	0.999802i	$-0.506335\pi$
$701$	−1.05383	−0.0398026	−0.0199013	+	0.999802i	$0.506335\pi$
$702$	0	0
$703$	12.5613	0.473758
$704$	0	0
$705$	−2.08793	−0.0786358
$706$	0	0
$707$	1.82744	0.0687281
$708$	0	0
$709$	−1.33736	−0.0502257	−0.0251129	−	0.999685i	$-0.507995\pi$
$709$	−1.33736	−0.0502257	−0.0251129	+	0.999685i	$0.507995\pi$
$710$	0	0
$711$	6.74411	0.252924
$712$	0	0
$713$	8.07437	0.302388
$714$	0	0
$715$	−11.3306	−0.423740
$716$	0	0
$717$	−27.8379	−1.03962
$718$	0	0
$719$	6.81276	0.254073	0.127037	−	0.991898i	$-0.459453\pi$
$719$	6.81276	0.254073	0.127037	+	0.991898i	$0.459453\pi$
$720$	0	0
$721$	5.63597	0.209894
$722$	0	0
$723$	−31.0024	−1.15299
$724$	0	0
$725$	−20.6409	−0.766582
$726$	0	0
$727$	20.6649	0.766419	0.383210	−	0.923661i	$-0.374819\pi$
$727$	20.6649	0.766419	0.383210	+	0.923661i	$0.374819\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.06522	−0.261317
$732$	0	0
$733$	−36.2674	−1.33957	−0.669784	−	0.742556i	$-0.733613\pi$
$733$	−36.2674	−1.33957	−0.669784	+	0.742556i	$0.733613\pi$
$734$	0	0
$735$	3.77185	0.139127
$736$	0	0
$737$	3.77872	0.139191
$738$	0	0
$739$	16.6388	0.612069	0.306034	−	0.952020i	$-0.400998\pi$
$739$	16.6388	0.612069	0.306034	+	0.952020i	$0.400998\pi$
$740$	0	0
$741$	8.61604	0.316518
$742$	0	0
$743$	4.04095	0.148248	0.0741241	−	0.997249i	$-0.476384\pi$
$743$	4.04095	0.148248	0.0741241	+	0.997249i	$0.476384\pi$
$744$	0	0
$745$	6.73508	0.246754
$746$	0	0
$747$	0.00272570	9.97281e−5 0
$748$	0	0
$749$	−7.02991	−0.256867
$750$	0	0
$751$	−25.2195	−0.920272	−0.460136	−	0.887849i	$-0.652199\pi$
$751$	−25.2195	−0.920272	−0.460136	+	0.887849i	$0.652199\pi$
$752$	0	0
$753$	2.88623	0.105180
$754$	0	0
$755$	−14.8177	−0.539272
$756$	0	0
$757$	22.8414	0.830184	0.415092	−	0.909779i	$-0.363749\pi$
$757$	22.8414	0.830184	0.415092	+	0.909779i	$0.363749\pi$
$758$	0	0
$759$	−14.6048	−0.530121
$760$	0	0
$761$	−30.8230	−1.11733	−0.558666	−	0.829392i	$-0.688687\pi$
$761$	−30.8230	−1.11733	−0.558666	+	0.829392i	$0.688687\pi$
$762$	0	0
$763$	8.76872	0.317449
$764$	0	0
$765$	−3.30546	−0.119509
$766$	0	0
$767$	−29.3657	−1.06033
$768$	0	0
$769$	−22.8669	−0.824600	−0.412300	−	0.911048i	$-0.635274\pi$
$769$	−22.8669	−0.824600	−0.412300	+	0.911048i	$0.635274\pi$
$770$	0	0
$771$	−10.5265	−0.379103
$772$	0	0
$773$	−31.4906	−1.13264	−0.566320	−	0.824186i	$-0.691633\pi$
$773$	−31.4906	−1.13264	−0.566320	+	0.824186i	$0.691633\pi$
$774$	0	0
$775$	15.9539	0.573081
$776$	0	0
$777$	4.07069	0.146035
$778$	0	0
$779$	−31.8576	−1.14142
$780$	0	0
$781$	−60.9317	−2.18031
$782$	0	0
$783$	4.47124	0.159789
$784$	0	0
$785$	−6.47848	−0.231227
$786$	0	0
$787$	−17.5858	−0.626866	−0.313433	−	0.949610i	$-0.601479\pi$
$787$	−17.5858	−0.626866	−0.313433	+	0.949610i	$0.601479\pi$
$788$	0	0
$789$	−13.9049	−0.495029
$790$	0	0
$791$	−12.3221	−0.438125
$792$	0	0
$793$	34.6751	1.23135
$794$	0	0
$795$	−3.54456	−0.125713
$796$	0	0
$797$	18.0559	0.639573	0.319787	−	0.947490i	$-0.396389\pi$
$797$	18.0559	0.639573	0.319787	+	0.947490i	$0.396389\pi$
$798$	0	0
$799$	17.9896	0.636427
$800$	0	0
$801$	9.85457	0.348194
$802$	0	0
$803$	−69.7955	−2.46303
$804$	0	0
$805$	1.38073	0.0486645
$806$	0	0
$807$	8.54423	0.300771
$808$	0	0
$809$	46.6237	1.63920	0.819600	−	0.572936i	$-0.194195\pi$
$809$	46.6237	1.63920	0.819600	+	0.572936i	$0.194195\pi$
$810$	0	0
$811$	17.3725	0.610030	0.305015	−	0.952348i	$-0.401339\pi$
$811$	17.3725	0.610030	0.305015	+	0.952348i	$0.401339\pi$
$812$	0	0
$813$	3.35225	0.117569
$814$	0	0
$815$	1.11358	0.0390071
$816$	0	0
$817$	−3.89788	−0.136370
$818$	0	0
$819$	2.79217	0.0975662
$820$	0	0
$821$	−6.23612	−0.217642	−0.108821	−	0.994061i	$-0.534708\pi$
$821$	−6.23612	−0.217642	−0.108821	+	0.994061i	$0.534708\pi$
$822$	0	0
$823$	−18.4109	−0.641762	−0.320881	−	0.947120i	$-0.603979\pi$
$823$	−18.4109	−0.641762	−0.320881	+	0.947120i	$0.603979\pi$
$824$	0	0
$825$	−28.8572	−1.00468
$826$	0	0
$827$	−2.54468	−0.0884873	−0.0442436	−	0.999021i	$-0.514088\pi$
$827$	−2.54468	−0.0884873	−0.0442436	+	0.999021i	$0.514088\pi$
$828$	0	0
$829$	40.5243	1.40747	0.703734	−	0.710463i	$-0.251515\pi$
$829$	40.5243	1.40747	0.703734	+	0.710463i	$0.251515\pi$
$830$	0	0
$831$	−11.5788	−0.401663
$832$	0	0
$833$	−32.4984	−1.12600
$834$	0	0
$835$	−0.619387	−0.0214348
$836$	0	0
$837$	−3.45595	−0.119455
$838$	0	0
$839$	21.9352	0.757287	0.378643	−	0.925543i	$-0.376391\pi$
$839$	21.9352	0.757287	0.378643	+	0.925543i	$0.376391\pi$
$840$	0	0
$841$	−9.00799	−0.310620
$842$	0	0
$843$	−7.10103	−0.244573
$844$	0	0
$845$	2.74767	0.0945228
$846$	0	0
$847$	26.7880	0.920445
$848$	0	0
$849$	−3.58868	−0.123163
$850$	0	0
$851$	−9.96789	−0.341695
$852$	0	0
$853$	7.93334	0.271632	0.135816	−	0.990734i	$-0.456634\pi$
$853$	7.93334	0.271632	0.135816	+	0.990734i	$0.456634\pi$
$854$	0	0
$855$	−1.82362	−0.0623666
$856$	0	0
$857$	−45.3773	−1.55006	−0.775029	−	0.631926i	$-0.782265\pi$
$857$	−45.3773	−1.55006	−0.775029	+	0.631926i	$0.782265\pi$
$858$	0	0
$859$	16.9407	0.578010	0.289005	−	0.957328i	$-0.406676\pi$
$859$	16.9407	0.578010	0.289005	+	0.957328i	$0.406676\pi$
$860$	0	0
$861$	−10.3240	−0.351839
$862$	0	0
$863$	−23.2827	−0.792551	−0.396275	−	0.918132i	$-0.629697\pi$
$863$	−23.2827	−0.792551	−0.396275	+	0.918132i	$0.629697\pi$
$864$	0	0
$865$	−3.93593	−0.133826
$866$	0	0
$867$	11.4799	0.389879
$868$	0	0
$869$	42.1579	1.43011
$870$	0	0
$871$	1.76899	0.0599400
$872$	0	0
$873$	−8.92479	−0.302058
$874$	0	0
$875$	5.68302	0.192121
$876$	0	0
$877$	32.5659	1.09967	0.549836	−	0.835272i	$-0.314690\pi$
$877$	32.5659	1.09967	0.549836	+	0.835272i	$0.314690\pi$
$878$	0	0
$879$	−23.4842	−0.792104
$880$	0	0
$881$	20.9764	0.706712	0.353356	−	0.935489i	$-0.385040\pi$
$881$	20.9764	0.706712	0.353356	+	0.935489i	$0.385040\pi$
$882$	0	0
$883$	30.1958	1.01617	0.508084	−	0.861307i	$-0.330354\pi$
$883$	30.1958	1.01617	0.508084	+	0.861307i	$0.330354\pi$
$884$	0	0
$885$	6.21537	0.208927
$886$	0	0
$887$	−37.7926	−1.26895	−0.634475	−	0.772943i	$-0.718784\pi$
$887$	−37.7926	−1.26895	−0.634475	+	0.772943i	$0.718784\pi$
$888$	0	0
$889$	8.85947	0.297137
$890$	0	0
$891$	6.25107	0.209419
$892$	0	0
$893$	9.92487	0.332123
$894$	0	0
$895$	−9.50042	−0.317564
$896$	0	0
$897$	−6.83718	−0.228287
$898$	0	0
$899$	−15.4524	−0.515366
$900$	0	0
$901$	30.5400	1.01744
$902$	0	0
$903$	−1.26317	−0.0420357
$904$	0	0
$905$	−1.82935	−0.0608098
$906$	0	0
$907$	−16.2201	−0.538581	−0.269291	−	0.963059i	$-0.586789\pi$
$907$	−16.2201	−0.538581	−0.269291	+	0.963059i	$0.586789\pi$
$908$	0	0
$909$	1.91531	0.0635267
$910$	0	0
$911$	−36.1700	−1.19837	−0.599183	−	0.800612i	$-0.704508\pi$
$911$	−36.1700	−1.19837	−0.599183	+	0.800612i	$0.704508\pi$
$912$	0	0
$913$	0.0170385	0.000563894 0
$914$	0	0
$915$	−7.33913	−0.242624
$916$	0	0
$917$	−2.14324	−0.0707759
$918$	0	0
$919$	46.1395	1.52200	0.761000	−	0.648752i	$-0.224709\pi$
$919$	46.1395	1.52200	0.761000	+	0.648752i	$0.224709\pi$
$920$	0	0
$921$	3.96744	0.130731
$922$	0	0
$923$	−28.5249	−0.938908
$924$	0	0
$925$	−19.6952	−0.647576
$926$	0	0
$927$	5.90694	0.194009
$928$	0	0
$929$	−38.7751	−1.27217	−0.636084	−	0.771619i	$-0.719447\pi$
$929$	−38.7751	−1.27217	−0.636084	+	0.771619i	$0.719447\pi$
$930$	0	0
$931$	−17.9293	−0.587610
$932$	0	0
$933$	1.69973	0.0556467
$934$	0	0
$935$	−20.6627	−0.675742
$936$	0	0
$937$	−11.0137	−0.359802	−0.179901	−	0.983685i	$-0.557578\pi$
$937$	−11.0137	−0.359802	−0.179901	+	0.983685i	$0.557578\pi$
$938$	0	0
$939$	9.65206	0.314983
$940$	0	0
$941$	59.1291	1.92755	0.963776	−	0.266713i	$-0.0859374\pi$
$941$	59.1291	1.92755	0.963776	+	0.266713i	$0.0859374\pi$
$942$	0	0
$943$	25.2803	0.823239
$944$	0	0
$945$	−0.590974	−0.0192244
$946$	0	0
$947$	−30.8243	−1.00165	−0.500827	−	0.865547i	$-0.666971\pi$
$947$	−30.8243	−1.00165	−0.500827	+	0.865547i	$0.666971\pi$
$948$	0	0
$949$	−32.6745	−1.06066
$950$	0	0
$951$	−8.58465	−0.278376
$952$	0	0
$953$	−18.3635	−0.594852	−0.297426	−	0.954745i	$-0.596128\pi$
$953$	−18.3635	−0.594852	−0.297426	+	0.954745i	$0.596128\pi$
$954$	0	0
$955$	−1.09486	−0.0354290
$956$	0	0
$957$	27.9501	0.903497
$958$	0	0
$959$	−0.852878	−0.0275409
$960$	0	0
$961$	−19.0564	−0.614723
$962$	0	0
$963$	−7.36790	−0.237427
$964$	0	0
$965$	6.97761	0.224617
$966$	0	0
$967$	−5.11727	−0.164560	−0.0822802	−	0.996609i	$-0.526220\pi$
$967$	−5.11727	−0.164560	−0.0822802	+	0.996609i	$0.526220\pi$
$968$	0	0
$969$	15.7124	0.504754
$970$	0	0
$971$	−56.0924	−1.80009	−0.900045	−	0.435797i	$-0.856467\pi$
$971$	−56.0924	−1.80009	−0.900045	+	0.435797i	$0.856467\pi$
$972$	0	0
$973$	1.06671	0.0341973
$974$	0	0
$975$	−13.5094	−0.432646
$976$	0	0
$977$	−9.10703	−0.291360	−0.145680	−	0.989332i	$-0.546537\pi$
$977$	−9.10703	−0.291360	−0.145680	+	0.989332i	$0.546537\pi$
$978$	0	0
$979$	61.6016	1.96880
$980$	0	0
$981$	9.19031	0.293424
$982$	0	0
$983$	−37.4710	−1.19514	−0.597570	−	0.801817i	$-0.703867\pi$
$983$	−37.4710	−1.19514	−0.597570	+	0.801817i	$0.703867\pi$
$984$	0	0
$985$	12.9690	0.413227
$986$	0	0
$987$	3.21631	0.102376
$988$	0	0
$989$	3.09313	0.0983558
$990$	0	0
$991$	−33.6068	−1.06756	−0.533778	−	0.845625i	$-0.679228\pi$
$991$	−33.6068	−1.06756	−0.533778	+	0.845625i	$0.679228\pi$
$992$	0	0
$993$	13.0207	0.413200
$994$	0	0
$995$	−7.94546	−0.251888
$996$	0	0
$997$	−10.8355	−0.343164	−0.171582	−	0.985170i	$-0.554888\pi$
$997$	−10.8355	−0.343164	−0.171582	+	0.985170i	$0.554888\pi$
$998$	0	0
$999$	4.26640	0.134983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.6	✓	12
4.3	odd	2		8016.2.a.bf.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.6	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.6		12	4.3	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.619387 q^{5} +0.954126 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.619387 q^{5} +0.954126 q^{7} +1.00000 q^{9} +6.25107 q^{11} +2.92641 q^{13} -0.619387 q^{15} +5.33666 q^{17} +2.94423 q^{19} +0.954126 q^{21} -2.33637 q^{23} -4.61636 q^{25} +1.00000 q^{27} +4.47124 q^{29} -3.45595 q^{31} +6.25107 q^{33} -0.590974 q^{35} +4.26640 q^{37} +2.92641 q^{39} -10.8203 q^{41} -1.32390 q^{43} -0.619387 q^{45} +3.37095 q^{47} -6.08964 q^{49} +5.33666 q^{51} +5.72269 q^{53} -3.87184 q^{55} +2.94423 q^{57} -10.0347 q^{59} +11.8490 q^{61} +0.954126 q^{63} -1.81258 q^{65} +0.604492 q^{67} -2.33637 q^{69} -9.74739 q^{71} -11.1654 q^{73} -4.61636 q^{75} +5.96431 q^{77} +6.74411 q^{79} +1.00000 q^{81} +0.00272570 q^{83} -3.30546 q^{85} +4.47124 q^{87} +9.85457 q^{89} +2.79217 q^{91} -3.45595 q^{93} -1.82362 q^{95} -8.92479 q^{97} +6.25107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more