LMFDB - Embedded newform 6008.2.a.b.1.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.32
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25524 q^{3} +4.39073 q^{5} -3.72993 q^{7} -1.42437 q^{9} +O(q^{10})$	$q+1.25524 q^{3} +4.39073 q^{5} -3.72993 q^{7} -1.42437 q^{9} +0.335194 q^{11} -6.63601 q^{13} +5.51143 q^{15} +5.58436 q^{17} -3.46817 q^{19} -4.68197 q^{21} -2.62801 q^{23} +14.2785 q^{25} -5.55365 q^{27} +4.43185 q^{29} -10.1044 q^{31} +0.420749 q^{33} -16.3771 q^{35} -4.76934 q^{37} -8.32979 q^{39} -5.06307 q^{41} -4.97138 q^{43} -6.25402 q^{45} +0.633362 q^{47} +6.91240 q^{49} +7.00973 q^{51} +6.19399 q^{53} +1.47175 q^{55} -4.35339 q^{57} +3.08291 q^{59} +7.40841 q^{61} +5.31280 q^{63} -29.1369 q^{65} -11.5319 q^{67} -3.29878 q^{69} -0.566138 q^{71} -12.8813 q^{73} +17.9230 q^{75} -1.25025 q^{77} +16.1760 q^{79} -2.69807 q^{81} +12.8071 q^{83} +24.5194 q^{85} +5.56304 q^{87} -7.99806 q^{89} +24.7519 q^{91} -12.6834 q^{93} -15.2278 q^{95} -14.3787 q^{97} -0.477440 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	1.25524	0.724714	0.362357	−	0.932039i	$-0.381972\pi$
$3$	1.25524	0.724714	0.362357	+	0.932039i	$0.381972\pi$
$4$	0	0
$5$	4.39073	1.96359	0.981797	−	0.189932i	$-0.0608268\pi$
$5$	4.39073	1.96359	0.981797	+	0.189932i	$0.0608268\pi$
$6$	0	0
$7$	−3.72993	−1.40978	−0.704891	−	0.709316i	$-0.749004\pi$
$7$	−3.72993	−1.40978	−0.704891	+	0.709316i	$0.749004\pi$
$8$	0	0
$9$	−1.42437	−0.474790
$10$	0	0
$11$	0.335194	0.101065	0.0505324	−	0.998722i	$-0.483908\pi$
$11$	0.335194	0.101065	0.0505324	+	0.998722i	$0.483908\pi$
$12$	0	0
$13$	−6.63601	−1.84050	−0.920248	−	0.391335i	$-0.872013\pi$
$13$	−6.63601	−1.84050	−0.920248	+	0.391335i	$0.872013\pi$
$14$	0	0
$15$	5.51143	1.42304
$16$	0	0
$17$	5.58436	1.35441	0.677204	−	0.735796i	$-0.263192\pi$
$17$	5.58436	1.35441	0.677204	+	0.735796i	$0.263192\pi$
$18$	0	0
$19$	−3.46817	−0.795652	−0.397826	−	0.917461i	$-0.630235\pi$
$19$	−3.46817	−0.795652	−0.397826	+	0.917461i	$0.630235\pi$
$20$	0	0
$21$	−4.68197	−1.02169
$22$	0	0
$23$	−2.62801	−0.547977	−0.273989	−	0.961733i	$-0.588343\pi$
$23$	−2.62801	−0.547977	−0.273989	+	0.961733i	$0.588343\pi$
$24$	0	0
$25$	14.2785	2.85570
$26$	0	0
$27$	−5.55365	−1.06880
$28$	0	0
$29$	4.43185	0.822974	0.411487	−	0.911416i	$-0.365010\pi$
$29$	4.43185	0.822974	0.411487	+	0.911416i	$0.365010\pi$
$30$	0	0
$31$	−10.1044	−1.81480	−0.907398	−	0.420271i	$-0.861935\pi$
$31$	−10.1044	−1.81480	−0.907398	+	0.420271i	$0.861935\pi$
$32$	0	0
$33$	0.420749	0.0732430
$34$	0	0
$35$	−16.3771	−2.76824
$36$	0	0
$37$	−4.76934	−0.784075	−0.392037	−	0.919949i	$-0.628230\pi$
$37$	−4.76934	−0.784075	−0.392037	+	0.919949i	$0.628230\pi$
$38$	0	0
$39$	−8.32979	−1.33383
$40$	0	0
$41$	−5.06307	−0.790719	−0.395359	−	0.918527i	$-0.629380\pi$
$41$	−5.06307	−0.790719	−0.395359	+	0.918527i	$0.629380\pi$
$42$	0	0
$43$	−4.97138	−0.758129	−0.379065	−	0.925370i	$-0.623754\pi$
$43$	−4.97138	−0.758129	−0.379065	+	0.925370i	$0.623754\pi$
$44$	0	0
$45$	−6.25402	−0.932294
$46$	0	0
$47$	0.633362	0.0923854	0.0461927	−	0.998933i	$-0.485291\pi$
$47$	0.633362	0.0923854	0.0461927	+	0.998933i	$0.485291\pi$
$48$	0	0
$49$	6.91240	0.987486
$50$	0	0
$51$	7.00973	0.981558
$52$	0	0
$53$	6.19399	0.850810	0.425405	−	0.905003i	$-0.360132\pi$
$53$	6.19399	0.850810	0.425405	+	0.905003i	$0.360132\pi$
$54$	0	0
$55$	1.47175	0.198450
$56$	0	0
$57$	−4.35339	−0.576620
$58$	0	0
$59$	3.08291	0.401361	0.200680	−	0.979657i	$-0.435685\pi$
$59$	3.08291	0.401361	0.200680	+	0.979657i	$0.435685\pi$
$60$	0	0
$61$	7.40841	0.948549	0.474275	−	0.880377i	$-0.342710\pi$
$61$	7.40841	0.948549	0.474275	+	0.880377i	$0.342710\pi$
$62$	0	0
$63$	5.31280	0.669350
$64$	0	0
$65$	−29.1369	−3.61399
$66$	0	0
$67$	−11.5319	−1.40884	−0.704421	−	0.709783i	$-0.748793\pi$
$67$	−11.5319	−1.40884	−0.704421	+	0.709783i	$0.748793\pi$
$68$	0	0
$69$	−3.29878	−0.397127
$70$	0	0
$71$	−0.566138	−0.0671882	−0.0335941	−	0.999436i	$-0.510695\pi$
$71$	−0.566138	−0.0671882	−0.0335941	+	0.999436i	$0.510695\pi$
$72$	0	0
$73$	−12.8813	−1.50764	−0.753822	−	0.657078i	$-0.771792\pi$
$73$	−12.8813	−1.50764	−0.753822	+	0.657078i	$0.771792\pi$
$74$	0	0
$75$	17.9230	2.06957
$76$	0	0
$77$	−1.25025	−0.142479
$78$	0	0
$79$	16.1760	1.81994	0.909970	−	0.414675i	$-0.136105\pi$
$79$	16.1760	1.81994	0.909970	+	0.414675i	$0.136105\pi$
$80$	0	0
$81$	−2.69807	−0.299785
$82$	0	0
$83$	12.8071	1.40576	0.702881	−	0.711307i	$-0.251897\pi$
$83$	12.8071	1.40576	0.702881	+	0.711307i	$0.251897\pi$
$84$	0	0
$85$	24.5194	2.65951
$86$	0	0
$87$	5.56304	0.596421
$88$	0	0
$89$	−7.99806	−0.847793	−0.423896	−	0.905711i	$-0.639338\pi$
$89$	−7.99806	−0.847793	−0.423896	+	0.905711i	$0.639338\pi$
$90$	0	0
$91$	24.7519	2.59470
$92$	0	0
$93$	−12.6834	−1.31521
$94$	0	0
$95$	−15.2278	−1.56234
$96$	0	0
$97$	−14.3787	−1.45994	−0.729970	−	0.683480i	$-0.760466\pi$
$97$	−14.3787	−1.45994	−0.729970	+	0.683480i	$0.760466\pi$
$98$	0	0
$99$	−0.477440	−0.0479845
$100$	0	0
$101$	12.0514	1.19916	0.599579	−	0.800316i	$-0.295335\pi$
$101$	12.0514	1.19916	0.599579	+	0.800316i	$0.295335\pi$
$102$	0	0
$103$	−8.74140	−0.861316	−0.430658	−	0.902515i	$-0.641718\pi$
$103$	−8.74140	−0.861316	−0.430658	+	0.902515i	$0.641718\pi$
$104$	0	0
$105$	−20.5573	−2.00618
$106$	0	0
$107$	−15.2038	−1.46981	−0.734903	−	0.678172i	$-0.762772\pi$
$107$	−15.2038	−1.46981	−0.734903	+	0.678172i	$0.762772\pi$
$108$	0	0
$109$	−10.3346	−0.989876	−0.494938	−	0.868928i	$-0.664809\pi$
$109$	−10.3346	−0.989876	−0.494938	+	0.868928i	$0.664809\pi$
$110$	0	0
$111$	−5.98667	−0.568230
$112$	0	0
$113$	−19.1314	−1.79973	−0.899867	−	0.436165i	$-0.856336\pi$
$113$	−19.1314	−1.79973	−0.899867	+	0.436165i	$0.856336\pi$
$114$	0	0
$115$	−11.5389	−1.07600
$116$	0	0
$117$	9.45212	0.873849
$118$	0	0
$119$	−20.8293	−1.90942
$120$	0	0
$121$	−10.8876	−0.989786
$122$	0	0
$123$	−6.35538	−0.573045
$124$	0	0
$125$	40.7395	3.64385
$126$	0	0
$127$	−6.76390	−0.600200	−0.300100	−	0.953908i	$-0.597020\pi$
$127$	−6.76390	−0.600200	−0.300100	+	0.953908i	$0.597020\pi$
$128$	0	0
$129$	−6.24029	−0.549427
$130$	0	0
$131$	−18.0192	−1.57435	−0.787173	−	0.616732i	$-0.788456\pi$
$131$	−18.0192	−1.57435	−0.787173	+	0.616732i	$0.788456\pi$
$132$	0	0
$133$	12.9360	1.12170
$134$	0	0
$135$	−24.3846	−2.09869
$136$	0	0
$137$	3.41751	0.291977	0.145989	−	0.989286i	$-0.453364\pi$
$137$	3.41751	0.291977	0.145989	+	0.989286i	$0.453364\pi$
$138$	0	0
$139$	1.26617	0.107395	0.0536977	−	0.998557i	$-0.482899\pi$
$139$	1.26617	0.107395	0.0536977	+	0.998557i	$0.482899\pi$
$140$	0	0
$141$	0.795023	0.0669530
$142$	0	0
$143$	−2.22435	−0.186009
$144$	0	0
$145$	19.4591	1.61599
$146$	0	0
$147$	8.67673	0.715645
$148$	0	0
$149$	−12.5806	−1.03065	−0.515323	−	0.856996i	$-0.672328\pi$
$149$	−12.5806	−1.03065	−0.515323	+	0.856996i	$0.672328\pi$
$150$	0	0
$151$	9.51437	0.774269	0.387134	−	0.922023i	$-0.373465\pi$
$151$	9.51437	0.774269	0.387134	+	0.922023i	$0.373465\pi$
$152$	0	0
$153$	−7.95420	−0.643059
$154$	0	0
$155$	−44.3655	−3.56353
$156$	0	0
$157$	−16.8830	−1.34741	−0.673706	−	0.739000i	$-0.735298\pi$
$157$	−16.8830	−1.34741	−0.673706	+	0.739000i	$0.735298\pi$
$158$	0	0
$159$	7.77495	0.616594
$160$	0	0
$161$	9.80228	0.772528
$162$	0	0
$163$	−4.38273	−0.343282	−0.171641	−	0.985160i	$-0.554907\pi$
$163$	−4.38273	−0.343282	−0.171641	+	0.985160i	$0.554907\pi$
$164$	0	0
$165$	1.84740	0.143820
$166$	0	0
$167$	2.79764	0.216488	0.108244	−	0.994124i	$-0.465477\pi$
$167$	2.79764	0.216488	0.108244	+	0.994124i	$0.465477\pi$
$168$	0	0
$169$	31.0366	2.38743
$170$	0	0
$171$	4.93995	0.377767
$172$	0	0
$173$	−6.40593	−0.487034	−0.243517	−	0.969897i	$-0.578301\pi$
$173$	−6.40593	−0.487034	−0.243517	+	0.969897i	$0.578301\pi$
$174$	0	0
$175$	−53.2579	−4.02592
$176$	0	0
$177$	3.86980	0.290872
$178$	0	0
$179$	−14.3346	−1.07142	−0.535710	−	0.844402i	$-0.679956\pi$
$179$	−14.3346	−1.07142	−0.535710	+	0.844402i	$0.679956\pi$
$180$	0	0
$181$	3.98508	0.296209	0.148104	−	0.988972i	$-0.452683\pi$
$181$	3.98508	0.296209	0.148104	+	0.988972i	$0.452683\pi$
$182$	0	0
$183$	9.29934	0.687427
$184$	0	0
$185$	−20.9409	−1.53961
$186$	0	0
$187$	1.87184	0.136883
$188$	0	0
$189$	20.7147	1.50678
$190$	0	0
$191$	13.5415	0.979829	0.489914	−	0.871771i	$-0.337028\pi$
$191$	13.5415	0.979829	0.489914	+	0.871771i	$0.337028\pi$
$192$	0	0
$193$	9.04622	0.651161	0.325581	−	0.945514i	$-0.394440\pi$
$193$	9.04622	0.651161	0.325581	+	0.945514i	$0.394440\pi$
$194$	0	0
$195$	−36.5739	−2.61911
$196$	0	0
$197$	10.5277	0.750069	0.375035	−	0.927011i	$-0.377631\pi$
$197$	10.5277	0.750069	0.375035	+	0.927011i	$0.377631\pi$
$198$	0	0
$199$	−3.35669	−0.237949	−0.118975	−	0.992897i	$-0.537961\pi$
$199$	−3.35669	−0.237949	−0.118975	+	0.992897i	$0.537961\pi$
$200$	0	0
$201$	−14.4753	−1.02101
$202$	0	0
$203$	−16.5305	−1.16021
$204$	0	0
$205$	−22.2306	−1.55265
$206$	0	0
$207$	3.74325	0.260174
$208$	0	0
$209$	−1.16251	−0.0804124
$210$	0	0
$211$	22.4356	1.54453	0.772267	−	0.635299i	$-0.219123\pi$
$211$	22.4356	1.54453	0.772267	+	0.635299i	$0.219123\pi$
$212$	0	0
$213$	−0.710639	−0.0486922
$214$	0	0
$215$	−21.8280	−1.48866
$216$	0	0
$217$	37.6886	2.55847
$218$	0	0
$219$	−16.1692	−1.09261
$220$	0	0
$221$	−37.0579	−2.49278
$222$	0	0
$223$	13.3050	0.890969	0.445485	−	0.895290i	$-0.353031\pi$
$223$	13.3050	0.890969	0.445485	+	0.895290i	$0.353031\pi$
$224$	0	0
$225$	−20.3379	−1.35586
$226$	0	0
$227$	6.53166	0.433521	0.216761	−	0.976225i	$-0.430451\pi$
$227$	6.53166	0.433521	0.216761	+	0.976225i	$0.430451\pi$
$228$	0	0
$229$	23.0806	1.52521	0.762606	−	0.646863i	$-0.223920\pi$
$229$	23.0806	1.52521	0.762606	+	0.646863i	$0.223920\pi$
$230$	0	0
$231$	−1.56937	−0.103257
$232$	0	0
$233$	−1.30377	−0.0854126	−0.0427063	−	0.999088i	$-0.513598\pi$
$233$	−1.30377	−0.0854126	−0.0427063	+	0.999088i	$0.513598\pi$
$234$	0	0
$235$	2.78092	0.181407
$236$	0	0
$237$	20.3048	1.31894
$238$	0	0
$239$	6.30117	0.407589	0.203794	−	0.979014i	$-0.434673\pi$
$239$	6.30117	0.407589	0.203794	+	0.979014i	$0.434673\pi$
$240$	0	0
$241$	9.47972	0.610643	0.305321	−	0.952249i	$-0.401236\pi$
$241$	9.47972	0.610643	0.305321	+	0.952249i	$0.401236\pi$
$242$	0	0
$243$	13.2742	0.851542
$244$	0	0
$245$	30.3505	1.93902
$246$	0	0
$247$	23.0148	1.46440
$248$	0	0
$249$	16.0760	1.01878
$250$	0	0
$251$	−19.0411	−1.20186	−0.600931	−	0.799301i	$-0.705203\pi$
$251$	−19.0411	−1.20186	−0.600931	+	0.799301i	$0.705203\pi$
$252$	0	0
$253$	−0.880891	−0.0553812
$254$	0	0
$255$	30.7778	1.92738
$256$	0	0
$257$	7.99855	0.498936	0.249468	−	0.968383i	$-0.419744\pi$
$257$	7.99855	0.498936	0.249468	+	0.968383i	$0.419744\pi$
$258$	0	0
$259$	17.7893	1.10537
$260$	0	0
$261$	−6.31259	−0.390740
$262$	0	0
$263$	−5.68627	−0.350630	−0.175315	−	0.984512i	$-0.556094\pi$
$263$	−5.68627	−0.350630	−0.175315	+	0.984512i	$0.556094\pi$
$264$	0	0
$265$	27.1961	1.67065
$266$	0	0
$267$	−10.0395	−0.614407
$268$	0	0
$269$	−5.42860	−0.330987	−0.165494	−	0.986211i	$-0.552922\pi$
$269$	−5.42860	−0.330987	−0.165494	+	0.986211i	$0.552922\pi$
$270$	0	0
$271$	−4.45134	−0.270400	−0.135200	−	0.990818i	$-0.543168\pi$
$271$	−4.45134	−0.270400	−0.135200	+	0.990818i	$0.543168\pi$
$272$	0	0
$273$	31.0696	1.88042
$274$	0	0
$275$	4.78607	0.288611
$276$	0	0
$277$	−0.594859	−0.0357416	−0.0178708	−	0.999840i	$-0.505689\pi$
$277$	−0.594859	−0.0357416	−0.0178708	+	0.999840i	$0.505689\pi$
$278$	0	0
$279$	14.3923	0.861647
$280$	0	0
$281$	20.2008	1.20508	0.602539	−	0.798089i	$-0.294156\pi$
$281$	20.2008	1.20508	0.602539	+	0.798089i	$0.294156\pi$
$282$	0	0
$283$	8.98521	0.534115	0.267058	−	0.963681i	$-0.413949\pi$
$283$	8.98521	0.534115	0.267058	+	0.963681i	$0.413949\pi$
$284$	0	0
$285$	−19.1146	−1.13225
$286$	0	0
$287$	18.8849	1.11474
$288$	0	0
$289$	14.1851	0.834419
$290$	0	0
$291$	−18.0488	−1.05804
$292$	0	0
$293$	−5.23411	−0.305780	−0.152890	−	0.988243i	$-0.548858\pi$
$293$	−5.23411	−0.305780	−0.152890	+	0.988243i	$0.548858\pi$
$294$	0	0
$295$	13.5362	0.788110
$296$	0	0
$297$	−1.86155	−0.108018
$298$	0	0
$299$	17.4395	1.00855
$300$	0	0
$301$	18.5429	1.06880
$302$	0	0
$303$	15.1274	0.869046
$304$	0	0
$305$	32.5283	1.86257
$306$	0	0
$307$	−31.7945	−1.81461	−0.907305	−	0.420473i	$-0.861864\pi$
$307$	−31.7945	−1.81461	−0.907305	+	0.420473i	$0.861864\pi$
$308$	0	0
$309$	−10.9726	−0.624208
$310$	0	0
$311$	3.58229	0.203133	0.101566	−	0.994829i	$-0.467615\pi$
$311$	3.58229	0.203133	0.101566	+	0.994829i	$0.467615\pi$
$312$	0	0
$313$	19.2869	1.09016	0.545080	−	0.838384i	$-0.316499\pi$
$313$	19.2869	1.09016	0.545080	+	0.838384i	$0.316499\pi$
$314$	0	0
$315$	23.3271	1.31433
$316$	0	0
$317$	−17.9167	−1.00630	−0.503151	−	0.864198i	$-0.667826\pi$
$317$	−17.9167	−1.00630	−0.503151	+	0.864198i	$0.667826\pi$
$318$	0	0
$319$	1.48553	0.0831737
$320$	0	0
$321$	−19.0844	−1.06519
$322$	0	0
$323$	−19.3675	−1.07764
$324$	0	0
$325$	−94.7523	−5.25591
$326$	0	0
$327$	−12.9724	−0.717377
$328$	0	0
$329$	−2.36240	−0.130243
$330$	0	0
$331$	−30.2449	−1.66241	−0.831204	−	0.555967i	$-0.812348\pi$
$331$	−30.2449	−1.66241	−0.831204	+	0.555967i	$0.812348\pi$
$332$	0	0
$333$	6.79330	0.372271
$334$	0	0
$335$	−50.6333	−2.76639
$336$	0	0
$337$	25.3476	1.38077	0.690387	−	0.723440i	$-0.257440\pi$
$337$	25.3476	1.38077	0.690387	+	0.723440i	$0.257440\pi$
$338$	0	0
$339$	−24.0146	−1.30429
$340$	0	0
$341$	−3.38692	−0.183412
$342$	0	0
$343$	0.326736	0.0176421
$344$	0	0
$345$	−14.4841	−0.779796
$346$	0	0
$347$	27.4348	1.47278	0.736389	−	0.676558i	$-0.236529\pi$
$347$	27.4348	1.47278	0.736389	+	0.676558i	$0.236529\pi$
$348$	0	0
$349$	−12.4475	−0.666297	−0.333149	−	0.942874i	$-0.608111\pi$
$349$	−12.4475	−0.666297	−0.333149	+	0.942874i	$0.608111\pi$
$350$	0	0
$351$	36.8541	1.96712
$352$	0	0
$353$	−28.5836	−1.52135	−0.760677	−	0.649131i	$-0.775133\pi$
$353$	−28.5836	−1.52135	−0.760677	+	0.649131i	$0.775133\pi$
$354$	0	0
$355$	−2.48576	−0.131930
$356$	0	0
$357$	−26.1458	−1.38378
$358$	0	0
$359$	3.34397	0.176488	0.0882439	−	0.996099i	$-0.471875\pi$
$359$	3.34397	0.176488	0.0882439	+	0.996099i	$0.471875\pi$
$360$	0	0
$361$	−6.97181	−0.366937
$362$	0	0
$363$	−13.6666	−0.717312
$364$	0	0
$365$	−56.5584	−2.96040
$366$	0	0
$367$	−12.5017	−0.652585	−0.326292	−	0.945269i	$-0.605799\pi$
$367$	−12.5017	−0.652585	−0.326292	+	0.945269i	$0.605799\pi$
$368$	0	0
$369$	7.21168	0.375425
$370$	0	0
$371$	−23.1032	−1.19946
$372$	0	0
$373$	29.8112	1.54357	0.771784	−	0.635885i	$-0.219365\pi$
$373$	29.8112	1.54357	0.771784	+	0.635885i	$0.219365\pi$
$374$	0	0
$375$	51.1379	2.64075
$376$	0	0
$377$	−29.4098	−1.51468
$378$	0	0
$379$	−0.518401	−0.0266285	−0.0133142	−	0.999911i	$-0.504238\pi$
$379$	−0.518401	−0.0266285	−0.0133142	+	0.999911i	$0.504238\pi$
$380$	0	0
$381$	−8.49033	−0.434973
$382$	0	0
$383$	−14.5407	−0.742993	−0.371496	−	0.928434i	$-0.621155\pi$
$383$	−14.5407	−0.742993	−0.371496	+	0.928434i	$0.621155\pi$
$384$	0	0
$385$	−5.48951	−0.279772
$386$	0	0
$387$	7.08109	0.359952
$388$	0	0
$389$	34.3902	1.74365	0.871826	−	0.489816i	$-0.162936\pi$
$389$	34.3902	1.74365	0.871826	+	0.489816i	$0.162936\pi$
$390$	0	0
$391$	−14.6757	−0.742184
$392$	0	0
$393$	−22.6185	−1.14095
$394$	0	0
$395$	71.0244	3.57362
$396$	0	0
$397$	20.1567	1.01164	0.505818	−	0.862640i	$-0.331191\pi$
$397$	20.1567	1.01164	0.505818	+	0.862640i	$0.331191\pi$
$398$	0	0
$399$	16.2378	0.812909
$400$	0	0
$401$	−22.3751	−1.11736	−0.558680	−	0.829383i	$-0.688692\pi$
$401$	−22.3751	−1.11736	−0.558680	+	0.829383i	$0.688692\pi$
$402$	0	0
$403$	67.0526	3.34013
$404$	0	0
$405$	−11.8465	−0.588656
$406$	0	0
$407$	−1.59865	−0.0792423
$408$	0	0
$409$	15.0462	0.743987	0.371994	−	0.928235i	$-0.378674\pi$
$409$	15.0462	0.743987	0.371994	+	0.928235i	$0.378674\pi$
$410$	0	0
$411$	4.28980	0.211600
$412$	0	0
$413$	−11.4990	−0.565831
$414$	0	0
$415$	56.2326	2.76035
$416$	0	0
$417$	1.58935	0.0778309
$418$	0	0
$419$	−21.6630	−1.05831	−0.529155	−	0.848525i	$-0.677491\pi$
$419$	−21.6630	−1.05831	−0.529155	+	0.848525i	$0.677491\pi$
$420$	0	0
$421$	−8.06331	−0.392982	−0.196491	−	0.980506i	$-0.562955\pi$
$421$	−8.06331	−0.392982	−0.196491	+	0.980506i	$0.562955\pi$
$422$	0	0
$423$	−0.902142	−0.0438636
$424$	0	0
$425$	79.7364	3.86779
$426$	0	0
$427$	−27.6329	−1.33725
$428$	0	0
$429$	−2.79209	−0.134804
$430$	0	0
$431$	24.3222	1.17156	0.585779	−	0.810471i	$-0.300789\pi$
$431$	24.3222	1.17156	0.585779	+	0.810471i	$0.300789\pi$
$432$	0	0
$433$	1.88103	0.0903966	0.0451983	−	0.998978i	$-0.485608\pi$
$433$	1.88103	0.0903966	0.0451983	+	0.998978i	$0.485608\pi$
$434$	0	0
$435$	24.4258	1.17113
$436$	0	0
$437$	9.11436	0.435999
$438$	0	0
$439$	−16.0473	−0.765898	−0.382949	−	0.923770i	$-0.625091\pi$
$439$	−16.0473	−0.765898	−0.382949	+	0.923770i	$0.625091\pi$
$440$	0	0
$441$	−9.84581	−0.468848
$442$	0	0
$443$	26.6310	1.26528	0.632639	−	0.774447i	$-0.281972\pi$
$443$	26.6310	1.26528	0.632639	+	0.774447i	$0.281972\pi$
$444$	0	0
$445$	−35.1173	−1.66472
$446$	0	0
$447$	−15.7917	−0.746923
$448$	0	0
$449$	−23.3422	−1.10159	−0.550794	−	0.834641i	$-0.685675\pi$
$449$	−23.3422	−1.10159	−0.550794	+	0.834641i	$0.685675\pi$
$450$	0	0
$451$	−1.69711	−0.0799138
$452$	0	0
$453$	11.9428	0.561123
$454$	0	0
$455$	108.679	5.09494
$456$	0	0
$457$	35.4364	1.65765	0.828823	−	0.559511i	$-0.189011\pi$
$457$	35.4364	1.65765	0.828823	+	0.559511i	$0.189011\pi$
$458$	0	0
$459$	−31.0136	−1.44759
$460$	0	0
$461$	25.4362	1.18468	0.592340	−	0.805688i	$-0.298204\pi$
$461$	25.4362	1.18468	0.592340	+	0.805688i	$0.298204\pi$
$462$	0	0
$463$	−8.72845	−0.405646	−0.202823	−	0.979215i	$-0.565012\pi$
$463$	−8.72845	−0.405646	−0.202823	+	0.979215i	$0.565012\pi$
$464$	0	0
$465$	−55.6895	−2.58254
$466$	0	0
$467$	42.7389	1.97772	0.988861	−	0.148843i	$-0.0475548\pi$
$467$	42.7389	1.97772	0.988861	+	0.148843i	$0.0475548\pi$
$468$	0	0
$469$	43.0131	1.98616
$470$	0	0
$471$	−21.1923	−0.976488
$472$	0	0
$473$	−1.66638	−0.0766201
$474$	0	0
$475$	−49.5203	−2.27215
$476$	0	0
$477$	−8.82252	−0.403956
$478$	0	0
$479$	−8.78288	−0.401300	−0.200650	−	0.979663i	$-0.564305\pi$
$479$	−8.78288	−0.401300	−0.200650	+	0.979663i	$0.564305\pi$
$480$	0	0
$481$	31.6494	1.44309
$482$	0	0
$483$	12.3042	0.559862
$484$	0	0
$485$	−63.1332	−2.86673
$486$	0	0
$487$	2.20490	0.0999133	0.0499567	−	0.998751i	$-0.484092\pi$
$487$	2.20490	0.0999133	0.0499567	+	0.998751i	$0.484092\pi$
$488$	0	0
$489$	−5.50138	−0.248781
$490$	0	0
$491$	−19.3931	−0.875200	−0.437600	−	0.899170i	$-0.644171\pi$
$491$	−19.3931	−0.875200	−0.437600	+	0.899170i	$0.644171\pi$
$492$	0	0
$493$	24.7491	1.11464
$494$	0	0
$495$	−2.09631	−0.0942221
$496$	0	0
$497$	2.11166	0.0947207
$498$	0	0
$499$	0.127065	0.00568823	0.00284412	−	0.999996i	$-0.499095\pi$
$499$	0.127065	0.00568823	0.00284412	+	0.999996i	$0.499095\pi$
$500$	0	0
$501$	3.51171	0.156892
$502$	0	0
$503$	−14.7666	−0.658412	−0.329206	−	0.944258i	$-0.606781\pi$
$503$	−14.7666	−0.658412	−0.329206	+	0.944258i	$0.606781\pi$
$504$	0	0
$505$	52.9144	2.35466
$506$	0	0
$507$	38.9584	1.73020
$508$	0	0
$509$	−9.40117	−0.416700	−0.208350	−	0.978054i	$-0.566809\pi$
$509$	−9.40117	−0.416700	−0.208350	+	0.978054i	$0.566809\pi$
$510$	0	0
$511$	48.0465	2.12545
$512$	0	0
$513$	19.2610	0.850394
$514$	0	0
$515$	−38.3811	−1.69128
$516$	0	0
$517$	0.212299	0.00933691
$518$	0	0
$519$	−8.04099	−0.352960
$520$	0	0
$521$	−31.7278	−1.39002	−0.695010	−	0.719000i	$-0.744600\pi$
$521$	−31.7278	−1.39002	−0.695010	+	0.719000i	$0.744600\pi$
$522$	0	0
$523$	15.9567	0.697736	0.348868	−	0.937172i	$-0.386566\pi$
$523$	15.9567	0.697736	0.348868	+	0.937172i	$0.386566\pi$
$524$	0	0
$525$	−66.8515	−2.91764
$526$	0	0
$527$	−56.4264	−2.45797
$528$	0	0
$529$	−16.0936	−0.699721
$530$	0	0
$531$	−4.39120	−0.190562
$532$	0	0
$533$	33.5986	1.45532
$534$	0	0
$535$	−66.7558	−2.88610
$536$	0	0
$537$	−17.9934	−0.776473
$538$	0	0
$539$	2.31699	0.0998000
$540$	0	0
$541$	31.9904	1.37538	0.687688	−	0.726007i	$-0.258626\pi$
$541$	31.9904	1.37538	0.687688	+	0.726007i	$0.258626\pi$
$542$	0	0
$543$	5.00224	0.214667
$544$	0	0
$545$	−45.3765	−1.94372
$546$	0	0
$547$	−29.3708	−1.25580	−0.627902	−	0.778292i	$-0.716086\pi$
$547$	−29.3708	−1.25580	−0.627902	+	0.778292i	$0.716086\pi$
$548$	0	0
$549$	−10.5523	−0.450361
$550$	0	0
$551$	−15.3704	−0.654801
$552$	0	0
$553$	−60.3353	−2.56572
$554$	0	0
$555$	−26.2859	−1.11577
$556$	0	0
$557$	20.3153	0.860785	0.430393	−	0.902642i	$-0.358375\pi$
$557$	20.3153	0.860785	0.430393	+	0.902642i	$0.358375\pi$
$558$	0	0
$559$	32.9901	1.39533
$560$	0	0
$561$	2.34962	0.0992009
$562$	0	0
$563$	22.8787	0.964223	0.482112	−	0.876110i	$-0.339870\pi$
$563$	22.8787	0.964223	0.482112	+	0.876110i	$0.339870\pi$
$564$	0	0
$565$	−84.0010	−3.53395
$566$	0	0
$567$	10.0636	0.422632
$568$	0	0
$569$	7.34976	0.308118	0.154059	−	0.988062i	$-0.450765\pi$
$569$	7.34976	0.308118	0.154059	+	0.988062i	$0.450765\pi$
$570$	0	0
$571$	33.7168	1.41100	0.705502	−	0.708707i	$-0.250721\pi$
$571$	33.7168	1.41100	0.705502	+	0.708707i	$0.250721\pi$
$572$	0	0
$573$	16.9979	0.710095
$574$	0	0
$575$	−37.5240	−1.56486
$576$	0	0
$577$	−15.0193	−0.625263	−0.312631	−	0.949875i	$-0.601210\pi$
$577$	−15.0193	−0.625263	−0.312631	+	0.949875i	$0.601210\pi$
$578$	0	0
$579$	11.3552	0.471906
$580$	0	0
$581$	−47.7696	−1.98182
$582$	0	0
$583$	2.07619	0.0859869
$584$	0	0
$585$	41.5017	1.71588
$586$	0	0
$587$	−14.8722	−0.613843	−0.306922	−	0.951735i	$-0.599299\pi$
$587$	−14.8722	−0.613843	−0.306922	+	0.951735i	$0.599299\pi$
$588$	0	0
$589$	35.0436	1.44395
$590$	0	0
$591$	13.2148	0.543586
$592$	0	0
$593$	−2.67847	−0.109992	−0.0549959	−	0.998487i	$-0.517515\pi$
$593$	−2.67847	−0.109992	−0.0549959	+	0.998487i	$0.517515\pi$
$594$	0	0
$595$	−91.4559	−3.74933
$596$	0	0
$597$	−4.21345	−0.172445
$598$	0	0
$599$	36.0046	1.47111	0.735554	−	0.677467i	$-0.236922\pi$
$599$	36.0046	1.47111	0.735554	+	0.677467i	$0.236922\pi$
$600$	0	0
$601$	28.4925	1.16223	0.581117	−	0.813820i	$-0.302616\pi$
$601$	28.4925	1.16223	0.581117	+	0.813820i	$0.302616\pi$
$602$	0	0
$603$	16.4256	0.668903
$604$	0	0
$605$	−47.8047	−1.94354
$606$	0	0
$607$	4.01590	0.163000	0.0815002	−	0.996673i	$-0.474029\pi$
$607$	4.01590	0.163000	0.0815002	+	0.996673i	$0.474029\pi$
$608$	0	0
$609$	−20.7498	−0.840823
$610$	0	0
$611$	−4.20300	−0.170035
$612$	0	0
$613$	−26.4257	−1.06732	−0.533662	−	0.845698i	$-0.679184\pi$
$613$	−26.4257	−1.06732	−0.533662	+	0.845698i	$0.679184\pi$
$614$	0	0
$615$	−27.9047	−1.12523
$616$	0	0
$617$	16.9602	0.682792	0.341396	−	0.939920i	$-0.389100\pi$
$617$	16.9602	0.682792	0.341396	+	0.939920i	$0.389100\pi$
$618$	0	0
$619$	−39.2046	−1.57577	−0.787883	−	0.615825i	$-0.788823\pi$
$619$	−39.2046	−1.57577	−0.787883	+	0.615825i	$0.788823\pi$
$620$	0	0
$621$	14.5950	0.585678
$622$	0	0
$623$	29.8322	1.19520
$624$	0	0
$625$	107.483	4.29934
$626$	0	0
$627$	−1.45923	−0.0582760
$628$	0	0
$629$	−26.6337	−1.06196
$630$	0	0
$631$	48.6081	1.93506	0.967529	−	0.252761i	$-0.0813387\pi$
$631$	48.6081	1.93506	0.967529	+	0.252761i	$0.0813387\pi$
$632$	0	0
$633$	28.1621	1.11934
$634$	0	0
$635$	−29.6985	−1.17855
$636$	0	0
$637$	−45.8707	−1.81746
$638$	0	0
$639$	0.806389	0.0319002
$640$	0	0
$641$	−13.1633	−0.519920	−0.259960	−	0.965619i	$-0.583709\pi$
$641$	−13.1633	−0.519920	−0.259960	+	0.965619i	$0.583709\pi$
$642$	0	0
$643$	−23.5489	−0.928677	−0.464339	−	0.885658i	$-0.653708\pi$
$643$	−23.5489	−0.928677	−0.464339	+	0.885658i	$0.653708\pi$
$644$	0	0
$645$	−27.3994	−1.07885
$646$	0	0
$647$	−8.32122	−0.327141	−0.163570	−	0.986532i	$-0.552301\pi$
$647$	−8.32122	−0.327141	−0.163570	+	0.986532i	$0.552301\pi$
$648$	0	0
$649$	1.03337	0.0405634
$650$	0	0
$651$	47.3083	1.85416
$652$	0	0
$653$	12.8063	0.501148	0.250574	−	0.968097i	$-0.419381\pi$
$653$	12.8063	0.501148	0.250574	+	0.968097i	$0.419381\pi$
$654$	0	0
$655$	−79.1176	−3.09138
$656$	0	0
$657$	18.3478	0.715814
$658$	0	0
$659$	−35.2512	−1.37319	−0.686595	−	0.727040i	$-0.740895\pi$
$659$	−35.2512	−1.37319	−0.686595	+	0.727040i	$0.740895\pi$
$660$	0	0
$661$	−0.199310	−0.00775224	−0.00387612	−	0.999992i	$-0.501234\pi$
$661$	−0.199310	−0.00775224	−0.00387612	+	0.999992i	$0.501234\pi$
$662$	0	0
$663$	−46.5166	−1.80655
$664$	0	0
$665$	56.7986	2.20256
$666$	0	0
$667$	−11.6469	−0.450971
$668$	0	0
$669$	16.7010	0.645698
$670$	0	0
$671$	2.48325	0.0958649
$672$	0	0
$673$	−17.4544	−0.672819	−0.336410	−	0.941716i	$-0.609213\pi$
$673$	−17.4544	−0.672819	−0.336410	+	0.941716i	$0.609213\pi$
$674$	0	0
$675$	−79.2979	−3.05218
$676$	0	0
$677$	21.8693	0.840505	0.420253	−	0.907407i	$-0.361941\pi$
$677$	21.8693	0.840505	0.420253	+	0.907407i	$0.361941\pi$
$678$	0	0
$679$	53.6317	2.05820
$680$	0	0
$681$	8.19881	0.314179
$682$	0	0
$683$	7.71128	0.295064	0.147532	−	0.989057i	$-0.452867\pi$
$683$	7.71128	0.295064	0.147532	+	0.989057i	$0.452867\pi$
$684$	0	0
$685$	15.0054	0.573325
$686$	0	0
$687$	28.9718	1.10534
$688$	0	0
$689$	−41.1033	−1.56591
$690$	0	0
$691$	−21.2904	−0.809926	−0.404963	−	0.914333i	$-0.632716\pi$
$691$	−21.2904	−0.809926	−0.404963	+	0.914333i	$0.632716\pi$
$692$	0	0
$693$	1.78082	0.0676477
$694$	0	0
$695$	5.55942	0.210881
$696$	0	0
$697$	−28.2740	−1.07096
$698$	0	0
$699$	−1.63654	−0.0618997
$700$	0	0
$701$	9.86278	0.372512	0.186256	−	0.982501i	$-0.440365\pi$
$701$	9.86278	0.372512	0.186256	+	0.982501i	$0.440365\pi$
$702$	0	0
$703$	16.5409	0.623851
$704$	0	0
$705$	3.49073	0.131469
$706$	0	0
$707$	−44.9509	−1.69055
$708$	0	0
$709$	−21.0510	−0.790586	−0.395293	−	0.918555i	$-0.629357\pi$
$709$	−21.0510	−0.790586	−0.395293	+	0.918555i	$0.629357\pi$
$710$	0	0
$711$	−23.0406	−0.864088
$712$	0	0
$713$	26.5543	0.994467
$714$	0	0
$715$	−9.76652	−0.365247
$716$	0	0
$717$	7.90948	0.295385
$718$	0	0
$719$	10.2812	0.383424	0.191712	−	0.981451i	$-0.438596\pi$
$719$	10.2812	0.383424	0.191712	+	0.981451i	$0.438596\pi$
$720$	0	0
$721$	32.6048	1.21427
$722$	0	0
$723$	11.8993	0.442541
$724$	0	0
$725$	63.2803	2.35017
$726$	0	0
$727$	10.9986	0.407915	0.203958	−	0.978980i	$-0.434620\pi$
$727$	10.9986	0.407915	0.203958	+	0.978980i	$0.434620\pi$
$728$	0	0
$729$	24.7566	0.916910
$730$	0	0
$731$	−27.7620	−1.02682
$732$	0	0
$733$	−49.8169	−1.84003	−0.920015	−	0.391882i	$-0.871824\pi$
$733$	−49.8169	−1.84003	−0.920015	+	0.391882i	$0.871824\pi$
$734$	0	0
$735$	38.0972	1.40524
$736$	0	0
$737$	−3.86541	−0.142384
$738$	0	0
$739$	−10.6389	−0.391357	−0.195678	−	0.980668i	$-0.562691\pi$
$739$	−10.6389	−0.391357	−0.195678	+	0.980668i	$0.562691\pi$
$740$	0	0
$741$	28.8891	1.06127
$742$	0	0
$743$	−52.1630	−1.91367	−0.956837	−	0.290625i	$-0.906137\pi$
$743$	−52.1630	−1.91367	−0.956837	+	0.290625i	$0.906137\pi$
$744$	0	0
$745$	−55.2382	−2.02377
$746$	0	0
$747$	−18.2420	−0.667441
$748$	0	0
$749$	56.7091	2.07211
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−23.9011	−0.871006
$754$	0	0
$755$	41.7751	1.52035
$756$	0	0
$757$	38.4968	1.39919	0.699595	−	0.714540i	$-0.253364\pi$
$757$	38.4968	1.39919	0.699595	+	0.714540i	$0.253364\pi$
$758$	0	0
$759$	−1.10573	−0.0401355
$760$	0	0
$761$	28.1326	1.01980	0.509902	−	0.860232i	$-0.329682\pi$
$761$	28.1326	1.01980	0.509902	+	0.860232i	$0.329682\pi$
$762$	0	0
$763$	38.5474	1.39551
$764$	0	0
$765$	−34.9247	−1.26271
$766$	0	0
$767$	−20.4582	−0.738703
$768$	0	0
$769$	−8.58509	−0.309586	−0.154793	−	0.987947i	$-0.549471\pi$
$769$	−8.58509	−0.309586	−0.154793	+	0.987947i	$0.549471\pi$
$770$	0	0
$771$	10.0401	0.361586
$772$	0	0
$773$	22.0473	0.792986	0.396493	−	0.918038i	$-0.370227\pi$
$773$	22.0473	0.792986	0.396493	+	0.918038i	$0.370227\pi$
$774$	0	0
$775$	−144.275	−5.18252
$776$	0	0
$777$	22.3299	0.801081
$778$	0	0
$779$	17.5596	0.629137
$780$	0	0
$781$	−0.189766	−0.00679036
$782$	0	0
$783$	−24.6130	−0.879595
$784$	0	0
$785$	−74.1288	−2.64577
$786$	0	0
$787$	−12.3972	−0.441911	−0.220955	−	0.975284i	$-0.570918\pi$
$787$	−12.3972	−0.441911	−0.220955	+	0.975284i	$0.570918\pi$
$788$	0	0
$789$	−7.13764	−0.254107
$790$	0	0
$791$	71.3590	2.53723
$792$	0	0
$793$	−49.1622	−1.74580
$794$	0	0
$795$	34.1377	1.21074
$796$	0	0
$797$	−15.0503	−0.533109	−0.266554	−	0.963820i	$-0.585885\pi$
$797$	−15.0503	−0.533109	−0.266554	+	0.963820i	$0.585885\pi$
$798$	0	0
$799$	3.53693	0.125127
$800$	0	0
$801$	11.3922	0.402523
$802$	0	0
$803$	−4.31774	−0.152370
$804$	0	0
$805$	43.0392	1.51693
$806$	0	0
$807$	−6.81420	−0.239871
$808$	0	0
$809$	−21.1326	−0.742984	−0.371492	−	0.928436i	$-0.621154\pi$
$809$	−21.1326	−0.742984	−0.371492	+	0.928436i	$0.621154\pi$
$810$	0	0
$811$	30.8794	1.08432	0.542162	−	0.840274i	$-0.317606\pi$
$811$	30.8794	1.08432	0.542162	+	0.840274i	$0.317606\pi$
$812$	0	0
$813$	−5.58750	−0.195962
$814$	0	0
$815$	−19.2434	−0.674066
$816$	0	0
$817$	17.2416	0.603207
$818$	0	0
$819$	−35.2558	−1.23194
$820$	0	0
$821$	−31.8193	−1.11050	−0.555251	−	0.831683i	$-0.687378\pi$
$821$	−31.8193	−1.11050	−0.555251	+	0.831683i	$0.687378\pi$
$822$	0	0
$823$	28.8273	1.00486	0.502428	−	0.864619i	$-0.332440\pi$
$823$	28.8273	1.00486	0.502428	+	0.864619i	$0.332440\pi$
$824$	0	0
$825$	6.00768	0.209160
$826$	0	0
$827$	−24.5278	−0.852915	−0.426458	−	0.904508i	$-0.640239\pi$
$827$	−24.5278	−0.852915	−0.426458	+	0.904508i	$0.640239\pi$
$828$	0	0
$829$	−35.4745	−1.23208	−0.616041	−	0.787714i	$-0.711264\pi$
$829$	−35.4745	−1.23208	−0.616041	+	0.787714i	$0.711264\pi$
$830$	0	0
$831$	−0.746691	−0.0259024
$832$	0	0
$833$	38.6014	1.33746
$834$	0	0
$835$	12.2837	0.425094
$836$	0	0
$837$	56.1161	1.93966
$838$	0	0
$839$	−9.39021	−0.324186	−0.162093	−	0.986775i	$-0.551824\pi$
$839$	−9.39021	−0.324186	−0.162093	+	0.986775i	$0.551824\pi$
$840$	0	0
$841$	−9.35870	−0.322714
$842$	0	0
$843$	25.3569	0.873337
$844$	0	0
$845$	136.273	4.68794
$846$	0	0
$847$	40.6102	1.39538
$848$	0	0
$849$	11.2786	0.387081
$850$	0	0
$851$	12.5339	0.429655
$852$	0	0
$853$	−17.4390	−0.597100	−0.298550	−	0.954394i	$-0.596503\pi$
$853$	−17.4390	−0.597100	−0.298550	+	0.954394i	$0.596503\pi$
$854$	0	0
$855$	21.6900	0.741782
$856$	0	0
$857$	−32.4703	−1.10917	−0.554583	−	0.832129i	$-0.687122\pi$
$857$	−32.4703	−1.10917	−0.554583	+	0.832129i	$0.687122\pi$
$858$	0	0
$859$	23.8328	0.813165	0.406582	−	0.913614i	$-0.366720\pi$
$859$	23.8328	0.813165	0.406582	+	0.913614i	$0.366720\pi$
$860$	0	0
$861$	23.7051	0.807869
$862$	0	0
$863$	40.1305	1.36606	0.683029	−	0.730392i	$-0.260662\pi$
$863$	40.1305	1.36606	0.683029	+	0.730392i	$0.260662\pi$
$864$	0	0
$865$	−28.1267	−0.956337
$866$	0	0
$867$	17.8058	0.604715
$868$	0	0
$869$	5.42209	0.183932
$870$	0	0
$871$	76.5255	2.59297
$872$	0	0
$873$	20.4806	0.693164
$874$	0	0
$875$	−151.955	−5.13703
$876$	0	0
$877$	−24.3447	−0.822060	−0.411030	−	0.911622i	$-0.634831\pi$
$877$	−24.3447	−0.822060	−0.411030	+	0.911622i	$0.634831\pi$
$878$	0	0
$879$	−6.57007	−0.221603
$880$	0	0
$881$	−27.2359	−0.917601	−0.458800	−	0.888539i	$-0.651721\pi$
$881$	−27.2359	−0.917601	−0.458800	+	0.888539i	$0.651721\pi$
$882$	0	0
$883$	−5.21691	−0.175563	−0.0877815	−	0.996140i	$-0.527978\pi$
$883$	−5.21691	−0.175563	−0.0877815	+	0.996140i	$0.527978\pi$
$884$	0	0
$885$	16.9912	0.571154
$886$	0	0
$887$	41.7894	1.40315	0.701576	−	0.712595i	$-0.252480\pi$
$887$	41.7894	1.40315	0.701576	+	0.712595i	$0.252480\pi$
$888$	0	0
$889$	25.2289	0.846151
$890$	0	0
$891$	−0.904375	−0.0302977
$892$	0	0
$893$	−2.19661	−0.0735066
$894$	0	0
$895$	−62.9394	−2.10383
$896$	0	0
$897$	21.8907	0.730910
$898$	0	0
$899$	−44.7810	−1.49353
$900$	0	0
$901$	34.5895	1.15234
$902$	0	0
$903$	23.2759	0.774572
$904$	0	0
$905$	17.4974	0.581634
$906$	0	0
$907$	10.8587	0.360558	0.180279	−	0.983615i	$-0.442300\pi$
$907$	10.8587	0.360558	0.180279	+	0.983615i	$0.442300\pi$
$908$	0	0
$909$	−17.1656	−0.569348
$910$	0	0
$911$	−2.84690	−0.0943221	−0.0471611	−	0.998887i	$-0.515017\pi$
$911$	−2.84690	−0.0943221	−0.0471611	+	0.998887i	$0.515017\pi$
$912$	0	0
$913$	4.29286	0.142073
$914$	0	0
$915$	40.8309	1.34983
$916$	0	0
$917$	67.2105	2.21949
$918$	0	0
$919$	−29.7791	−0.982321	−0.491161	−	0.871069i	$-0.663427\pi$
$919$	−29.7791	−0.982321	−0.491161	+	0.871069i	$0.663427\pi$
$920$	0	0
$921$	−39.9098	−1.31507
$922$	0	0
$923$	3.75689	0.123660
$924$	0	0
$925$	−68.0991	−2.23909
$926$	0	0
$927$	12.4510	0.408944
$928$	0	0
$929$	36.5135	1.19797	0.598984	−	0.800761i	$-0.295571\pi$
$929$	36.5135	1.19797	0.598984	+	0.800761i	$0.295571\pi$
$930$	0	0
$931$	−23.9734	−0.785696
$932$	0	0
$933$	4.49663	0.147213
$934$	0	0
$935$	8.21877	0.268782
$936$	0	0
$937$	−36.3632	−1.18793	−0.593967	−	0.804489i	$-0.702439\pi$
$937$	−36.3632	−1.18793	−0.593967	+	0.804489i	$0.702439\pi$
$938$	0	0
$939$	24.2097	0.790054
$940$	0	0
$941$	10.3357	0.336935	0.168467	−	0.985707i	$-0.446118\pi$
$941$	10.3357	0.336935	0.168467	+	0.985707i	$0.446118\pi$
$942$	0	0
$943$	13.3058	0.433296
$944$	0	0
$945$	90.9529	2.95870
$946$	0	0
$947$	−42.3939	−1.37762	−0.688808	−	0.724943i	$-0.741866\pi$
$947$	−42.3939	−1.37762	−0.688808	+	0.724943i	$0.741866\pi$
$948$	0	0
$949$	85.4805	2.77481
$950$	0	0
$951$	−22.4898	−0.729281
$952$	0	0
$953$	28.2563	0.915310	0.457655	−	0.889130i	$-0.348689\pi$
$953$	28.2563	0.915310	0.457655	+	0.889130i	$0.348689\pi$
$954$	0	0
$955$	59.4571	1.92399
$956$	0	0
$957$	1.86470	0.0602771
$958$	0	0
$959$	−12.7471	−0.411625
$960$	0	0
$961$	71.0981	2.29349
$962$	0	0
$963$	21.6558	0.697849
$964$	0	0
$965$	39.7195	1.27862
$966$	0	0
$967$	−29.4948	−0.948488	−0.474244	−	0.880393i	$-0.657279\pi$
$967$	−29.4948	−0.948488	−0.474244	+	0.880393i	$0.657279\pi$
$968$	0	0
$969$	−24.3109	−0.780979
$970$	0	0
$971$	24.5156	0.786743	0.393371	−	0.919380i	$-0.371309\pi$
$971$	24.5156	0.786743	0.393371	+	0.919380i	$0.371309\pi$
$972$	0	0
$973$	−4.72274	−0.151404
$974$	0	0
$975$	−118.937	−3.80903
$976$	0	0
$977$	−3.71085	−0.118721	−0.0593603	−	0.998237i	$-0.518906\pi$
$977$	−3.71085	−0.118721	−0.0593603	+	0.998237i	$0.518906\pi$
$978$	0	0
$979$	−2.68090	−0.0856820
$980$	0	0
$981$	14.7203	0.469983
$982$	0	0
$983$	18.8140	0.600072	0.300036	−	0.953928i	$-0.403001\pi$
$983$	18.8140	0.600072	0.300036	+	0.953928i	$0.403001\pi$
$984$	0	0
$985$	46.2244	1.47283
$986$	0	0
$987$	−2.96538	−0.0943891
$988$	0	0
$989$	13.0648	0.415437
$990$	0	0
$991$	−35.7069	−1.13427	−0.567133	−	0.823626i	$-0.691948\pi$
$991$	−35.7069	−1.13427	−0.567133	+	0.823626i	$0.691948\pi$
$992$	0	0
$993$	−37.9646	−1.20477
$994$	0	0
$995$	−14.7383	−0.467236
$996$	0	0
$997$	−39.4721	−1.25010	−0.625048	−	0.780587i	$-0.714920\pi$
$997$	−39.4721	−1.25010	−0.625048	+	0.780587i	$0.714920\pi$
$998$	0	0
$999$	26.4873	0.838020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.32	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.32	✓	44	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.25524 q^{3} +4.39073 q^{5} -3.72993 q^{7} -1.42437 q^{9} +O(q^{10})\)	\(q+1.25524 q^{3} +4.39073 q^{5} -3.72993 q^{7} -1.42437 q^{9} +0.335194 q^{11} -6.63601 q^{13} +5.51143 q^{15} +5.58436 q^{17} -3.46817 q^{19} -4.68197 q^{21} -2.62801 q^{23} +14.2785 q^{25} -5.55365 q^{27} +4.43185 q^{29} -10.1044 q^{31} +0.420749 q^{33} -16.3771 q^{35} -4.76934 q^{37} -8.32979 q^{39} -5.06307 q^{41} -4.97138 q^{43} -6.25402 q^{45} +0.633362 q^{47} +6.91240 q^{49} +7.00973 q^{51} +6.19399 q^{53} +1.47175 q^{55} -4.35339 q^{57} +3.08291 q^{59} +7.40841 q^{61} +5.31280 q^{63} -29.1369 q^{65} -11.5319 q^{67} -3.29878 q^{69} -0.566138 q^{71} -12.8813 q^{73} +17.9230 q^{75} -1.25025 q^{77} +16.1760 q^{79} -2.69807 q^{81} +12.8071 q^{83} +24.5194 q^{85} +5.56304 q^{87} -7.99806 q^{89} +24.7519 q^{91} -12.6834 q^{93} -15.2278 q^{95} -14.3787 q^{97} -0.477440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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