LMFDB - Embedded newform 6008.2.a.b.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.46379 q^{3} +1.54615 q^{5} -4.51355 q^{7} +3.07028 q^{9} +O(q^{10})$	$q-2.46379 q^{3} +1.54615 q^{5} -4.51355 q^{7} +3.07028 q^{9} -1.39622 q^{11} -1.88688 q^{13} -3.80940 q^{15} +5.81705 q^{17} -5.42202 q^{19} +11.1205 q^{21} +1.48892 q^{23} -2.60942 q^{25} -0.173161 q^{27} +6.06659 q^{29} +2.12645 q^{31} +3.43999 q^{33} -6.97863 q^{35} +5.90324 q^{37} +4.64888 q^{39} +0.00128388 q^{41} +1.74065 q^{43} +4.74712 q^{45} -4.47451 q^{47} +13.3722 q^{49} -14.3320 q^{51} +1.11696 q^{53} -2.15876 q^{55} +13.3587 q^{57} +3.60544 q^{59} -7.40424 q^{61} -13.8579 q^{63} -2.91740 q^{65} +10.7106 q^{67} -3.66840 q^{69} +6.58932 q^{71} -8.24164 q^{73} +6.42907 q^{75} +6.30190 q^{77} +7.11435 q^{79} -8.78421 q^{81} +3.52854 q^{83} +8.99404 q^{85} -14.9468 q^{87} -10.6326 q^{89} +8.51653 q^{91} -5.23914 q^{93} -8.38325 q^{95} +15.2081 q^{97} -4.28678 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.46379	−1.42247	−0.711236	−	0.702953i	$-0.751864\pi$
$3$	−2.46379	−1.42247	−0.711236	+	0.702953i	$0.751864\pi$
$4$	0	0
$5$	1.54615	0.691460	0.345730	−	0.938334i	$-0.387631\pi$
$5$	1.54615	0.691460	0.345730	+	0.938334i	$0.387631\pi$
$6$	0	0
$7$	−4.51355	−1.70596	−0.852981	−	0.521941i	$-0.825208\pi$
$7$	−4.51355	−1.70596	−0.852981	+	0.521941i	$0.825208\pi$
$8$	0	0
$9$	3.07028	1.02343
$10$	0	0
$11$	−1.39622	−0.420975	−0.210488	−	0.977597i	$-0.567505\pi$
$11$	−1.39622	−0.420975	−0.210488	+	0.977597i	$0.567505\pi$
$12$	0	0
$13$	−1.88688	−0.523326	−0.261663	−	0.965159i	$-0.584271\pi$
$13$	−1.88688	−0.523326	−0.261663	+	0.965159i	$0.584271\pi$
$14$	0	0
$15$	−3.80940	−0.983582
$16$	0	0
$17$	5.81705	1.41084	0.705421	−	0.708788i	$-0.250758\pi$
$17$	5.81705	1.41084	0.705421	+	0.708788i	$0.250758\pi$
$18$	0	0
$19$	−5.42202	−1.24390	−0.621948	−	0.783059i	$-0.713658\pi$
$19$	−5.42202	−1.24390	−0.621948	+	0.783059i	$0.713658\pi$
$20$	0	0
$21$	11.1205	2.42668
$22$	0	0
$23$	1.48892	0.310462	0.155231	−	0.987878i	$-0.450388\pi$
$23$	1.48892	0.310462	0.155231	+	0.987878i	$0.450388\pi$
$24$	0	0
$25$	−2.60942	−0.521883
$26$	0	0
$27$	−0.173161	−0.0333249
$28$	0	0
$29$	6.06659	1.12654	0.563269	−	0.826274i	$-0.309543\pi$
$29$	6.06659	1.12654	0.563269	+	0.826274i	$0.309543\pi$
$30$	0	0
$31$	2.12645	0.381922	0.190961	−	0.981598i	$-0.438840\pi$
$31$	2.12645	0.381922	0.190961	+	0.981598i	$0.438840\pi$
$32$	0	0
$33$	3.43999	0.598826
$34$	0	0
$35$	−6.97863	−1.17960
$36$	0	0
$37$	5.90324	0.970487	0.485244	−	0.874379i	$-0.338731\pi$
$37$	5.90324	0.970487	0.485244	+	0.874379i	$0.338731\pi$
$38$	0	0
$39$	4.64888	0.744417
$40$	0	0
$41$	0.00128388	0.000200509 0	0.000100254	−	1.00000i	$-0.499968\pi$
$41$	0.00128388	0.000200509 0	0.000100254		1.00000i	$0.499968\pi$
$42$	0	0
$43$	1.74065	0.265446	0.132723	−	0.991153i	$-0.457628\pi$
$43$	1.74065	0.265446	0.132723	+	0.991153i	$0.457628\pi$
$44$	0	0
$45$	4.74712	0.707659
$46$	0	0
$47$	−4.47451	−0.652674	−0.326337	−	0.945253i	$-0.605814\pi$
$47$	−4.47451	−0.652674	−0.326337	+	0.945253i	$0.605814\pi$
$48$	0	0
$49$	13.3722	1.91031
$50$	0	0
$51$	−14.3320	−2.00688
$52$	0	0
$53$	1.11696	0.153426	0.0767132	−	0.997053i	$-0.475557\pi$
$53$	1.11696	0.153426	0.0767132	+	0.997053i	$0.475557\pi$
$54$	0	0
$55$	−2.15876	−0.291087
$56$	0	0
$57$	13.3587	1.76941
$58$	0	0
$59$	3.60544	0.469389	0.234694	−	0.972069i	$-0.424591\pi$
$59$	3.60544	0.469389	0.234694	+	0.972069i	$0.424591\pi$
$60$	0	0
$61$	−7.40424	−0.948016	−0.474008	−	0.880521i	$-0.657193\pi$
$61$	−7.40424	−0.948016	−0.474008	+	0.880521i	$0.657193\pi$
$62$	0	0
$63$	−13.8579	−1.74593
$64$	0	0
$65$	−2.91740	−0.361859
$66$	0	0
$67$	10.7106	1.30851	0.654254	−	0.756275i	$-0.272983\pi$
$67$	10.7106	1.30851	0.654254	+	0.756275i	$0.272983\pi$
$68$	0	0
$69$	−3.66840	−0.441623
$70$	0	0
$71$	6.58932	0.782008	0.391004	−	0.920389i	$-0.372128\pi$
$71$	6.58932	0.782008	0.391004	+	0.920389i	$0.372128\pi$
$72$	0	0
$73$	−8.24164	−0.964611	−0.482305	−	0.876003i	$-0.660200\pi$
$73$	−8.24164	−0.964611	−0.482305	+	0.876003i	$0.660200\pi$
$74$	0	0
$75$	6.42907	0.742365
$76$	0	0
$77$	6.30190	0.718168
$78$	0	0
$79$	7.11435	0.800427	0.400213	−	0.916422i	$-0.368936\pi$
$79$	7.11435	0.800427	0.400213	+	0.916422i	$0.368936\pi$
$80$	0	0
$81$	−8.78421	−0.976024
$82$	0	0
$83$	3.52854	0.387308	0.193654	−	0.981070i	$-0.437966\pi$
$83$	3.52854	0.387308	0.193654	+	0.981070i	$0.437966\pi$
$84$	0	0
$85$	8.99404	0.975541
$86$	0	0
$87$	−14.9468	−1.60247
$88$	0	0
$89$	−10.6326	−1.12706	−0.563529	−	0.826096i	$-0.690557\pi$
$89$	−10.6326	−1.12706	−0.563529	+	0.826096i	$0.690557\pi$
$90$	0	0
$91$	8.51653	0.892775
$92$	0	0
$93$	−5.23914	−0.543274
$94$	0	0
$95$	−8.38325	−0.860104
$96$	0	0
$97$	15.2081	1.54415	0.772074	−	0.635532i	$-0.219219\pi$
$97$	15.2081	1.54415	0.772074	+	0.635532i	$0.219219\pi$
$98$	0	0
$99$	−4.28678	−0.430838
$100$	0	0
$101$	−19.6441	−1.95466	−0.977332	−	0.211711i	$-0.932097\pi$
$101$	−19.6441	−1.95466	−0.977332	+	0.211711i	$0.932097\pi$
$102$	0	0
$103$	−1.27909	−0.126032	−0.0630161	−	0.998013i	$-0.520072\pi$
$103$	−1.27909	−0.126032	−0.0630161	+	0.998013i	$0.520072\pi$
$104$	0	0
$105$	17.1939	1.67795
$106$	0	0
$107$	−10.0003	−0.966764	−0.483382	−	0.875409i	$-0.660592\pi$
$107$	−10.0003	−0.966764	−0.483382	+	0.875409i	$0.660592\pi$
$108$	0	0
$109$	12.9577	1.24112	0.620560	−	0.784159i	$-0.286906\pi$
$109$	12.9577	1.24112	0.620560	+	0.784159i	$0.286906\pi$
$110$	0	0
$111$	−14.5444	−1.38049
$112$	0	0
$113$	6.64548	0.625154	0.312577	−	0.949892i	$-0.398808\pi$
$113$	6.64548	0.625154	0.312577	+	0.949892i	$0.398808\pi$
$114$	0	0
$115$	2.30210	0.214672
$116$	0	0
$117$	−5.79325	−0.535586
$118$	0	0
$119$	−26.2556	−2.40684
$120$	0	0
$121$	−9.05058	−0.822780
$122$	0	0
$123$	−0.00316322	−0.000285218 0
$124$	0	0
$125$	−11.7653	−1.05232
$126$	0	0
$127$	−8.26613	−0.733501	−0.366750	−	0.930319i	$-0.619530\pi$
$127$	−8.26613	−0.733501	−0.366750	+	0.930319i	$0.619530\pi$
$128$	0	0
$129$	−4.28860	−0.377590
$130$	0	0
$131$	20.4110	1.78332	0.891659	−	0.452707i	$-0.149542\pi$
$131$	20.4110	1.78332	0.891659	+	0.452707i	$0.149542\pi$
$132$	0	0
$133$	24.4726	2.12204
$134$	0	0
$135$	−0.267733	−0.0230428
$136$	0	0
$137$	−12.8555	−1.09832	−0.549158	−	0.835718i	$-0.685052\pi$
$137$	−12.8555	−1.09832	−0.549158	+	0.835718i	$0.685052\pi$
$138$	0	0
$139$	9.49073	0.804993	0.402497	−	0.915422i	$-0.368143\pi$
$139$	9.49073	0.804993	0.402497	+	0.915422i	$0.368143\pi$
$140$	0	0
$141$	11.0243	0.928411
$142$	0	0
$143$	2.63449	0.220307
$144$	0	0
$145$	9.37987	0.778956
$146$	0	0
$147$	−32.9463	−2.71736
$148$	0	0
$149$	−0.941426	−0.0771246	−0.0385623	−	0.999256i	$-0.512278\pi$
$149$	−0.941426	−0.0771246	−0.0385623	+	0.999256i	$0.512278\pi$
$150$	0	0
$151$	0.737818	0.0600428	0.0300214	−	0.999549i	$-0.490442\pi$
$151$	0.737818	0.0600428	0.0300214	+	0.999549i	$0.490442\pi$
$152$	0	0
$153$	17.8600	1.44389
$154$	0	0
$155$	3.28782	0.264084
$156$	0	0
$157$	−8.92195	−0.712049	−0.356025	−	0.934477i	$-0.615868\pi$
$157$	−8.92195	−0.712049	−0.356025	+	0.934477i	$0.615868\pi$
$158$	0	0
$159$	−2.75196	−0.218245
$160$	0	0
$161$	−6.72033	−0.529636
$162$	0	0
$163$	−19.9660	−1.56386	−0.781930	−	0.623366i	$-0.785765\pi$
$163$	−19.9660	−1.56386	−0.781930	+	0.623366i	$0.785765\pi$
$164$	0	0
$165$	5.31875	0.414064
$166$	0	0
$167$	−15.6345	−1.20984	−0.604918	−	0.796287i	$-0.706794\pi$
$167$	−15.6345	−1.20984	−0.604918	+	0.796287i	$0.706794\pi$
$168$	0	0
$169$	−9.43969	−0.726130
$170$	0	0
$171$	−16.6471	−1.27304
$172$	0	0
$173$	0.265640	0.0201962	0.0100981	−	0.999949i	$-0.496786\pi$
$173$	0.265640	0.0201962	0.0100981	+	0.999949i	$0.496786\pi$
$174$	0	0
$175$	11.7777	0.890314
$176$	0	0
$177$	−8.88307	−0.667693
$178$	0	0
$179$	10.7752	0.805377	0.402689	−	0.915337i	$-0.368076\pi$
$179$	10.7752	0.805377	0.402689	+	0.915337i	$0.368076\pi$
$180$	0	0
$181$	10.2154	0.759307	0.379653	−	0.925129i	$-0.376043\pi$
$181$	10.2154	0.759307	0.379653	+	0.925129i	$0.376043\pi$
$182$	0	0
$183$	18.2425	1.34853
$184$	0	0
$185$	9.12731	0.671053
$186$	0	0
$187$	−8.12187	−0.593930
$188$	0	0
$189$	0.781572	0.0568510
$190$	0	0
$191$	7.80245	0.564566	0.282283	−	0.959331i	$-0.408908\pi$
$191$	7.80245	0.564566	0.282283	+	0.959331i	$0.408908\pi$
$192$	0	0
$193$	22.5406	1.62251	0.811255	−	0.584692i	$-0.198785\pi$
$193$	22.5406	1.62251	0.811255	+	0.584692i	$0.198785\pi$
$194$	0	0
$195$	7.18787	0.514734
$196$	0	0
$197$	−6.00587	−0.427900	−0.213950	−	0.976845i	$-0.568633\pi$
$197$	−6.00587	−0.427900	−0.213950	+	0.976845i	$0.568633\pi$
$198$	0	0
$199$	9.88724	0.700888	0.350444	−	0.936584i	$-0.386031\pi$
$199$	9.88724	0.700888	0.350444	+	0.936584i	$0.386031\pi$
$200$	0	0
$201$	−26.3887	−1.86132
$202$	0	0
$203$	−27.3819	−1.92183
$204$	0	0
$205$	0.00198508	0.000138644 0
$206$	0	0
$207$	4.57141	0.317735
$208$	0	0
$209$	7.57031	0.523649
$210$	0	0
$211$	5.04377	0.347227	0.173614	−	0.984814i	$-0.444456\pi$
$211$	5.04377	0.347227	0.173614	+	0.984814i	$0.444456\pi$
$212$	0	0
$213$	−16.2347	−1.11238
$214$	0	0
$215$	2.69131	0.183546
$216$	0	0
$217$	−9.59786	−0.651545
$218$	0	0
$219$	20.3057	1.37213
$220$	0	0
$221$	−10.9761	−0.738330
$222$	0	0
$223$	−10.7691	−0.721151	−0.360575	−	0.932730i	$-0.617420\pi$
$223$	−10.7691	−0.721151	−0.360575	+	0.932730i	$0.617420\pi$
$224$	0	0
$225$	−8.01165	−0.534110
$226$	0	0
$227$	−28.4526	−1.88846	−0.944232	−	0.329280i	$-0.893194\pi$
$227$	−28.4526	−1.88846	−0.944232	+	0.329280i	$0.893194\pi$
$228$	0	0
$229$	−6.54581	−0.432559	−0.216280	−	0.976331i	$-0.569392\pi$
$229$	−6.54581	−0.432559	−0.216280	+	0.976331i	$0.569392\pi$
$230$	0	0
$231$	−15.5266	−1.02157
$232$	0	0
$233$	−13.6326	−0.893104	−0.446552	−	0.894758i	$-0.647348\pi$
$233$	−13.6326	−0.893104	−0.446552	+	0.894758i	$0.647348\pi$
$234$	0	0
$235$	−6.91826	−0.451298
$236$	0	0
$237$	−17.5283	−1.13858
$238$	0	0
$239$	8.68138	0.561552	0.280776	−	0.959773i	$-0.409408\pi$
$239$	8.68138	0.561552	0.280776	+	0.959773i	$0.409408\pi$
$240$	0	0
$241$	6.08055	0.391683	0.195841	−	0.980636i	$-0.437256\pi$
$241$	6.08055	0.391683	0.195841	+	0.980636i	$0.437256\pi$
$242$	0	0
$243$	22.1620	1.42169
$244$	0	0
$245$	20.6754	1.32090
$246$	0	0
$247$	10.2307	0.650963
$248$	0	0
$249$	−8.69360	−0.550934
$250$	0	0
$251$	1.74489	0.110137	0.0550683	−	0.998483i	$-0.482462\pi$
$251$	1.74489	0.110137	0.0550683	+	0.998483i	$0.482462\pi$
$252$	0	0
$253$	−2.07886	−0.130697
$254$	0	0
$255$	−22.1595	−1.38768
$256$	0	0
$257$	11.8713	0.740513	0.370257	−	0.928930i	$-0.379270\pi$
$257$	11.8713	0.740513	0.370257	+	0.928930i	$0.379270\pi$
$258$	0	0
$259$	−26.6446	−1.65561
$260$	0	0
$261$	18.6262	1.15293
$262$	0	0
$263$	11.0704	0.682628	0.341314	−	0.939949i	$-0.389128\pi$
$263$	11.0704	0.682628	0.341314	+	0.939949i	$0.389128\pi$
$264$	0	0
$265$	1.72699	0.106088
$266$	0	0
$267$	26.1967	1.60321
$268$	0	0
$269$	14.1665	0.863747	0.431873	−	0.901934i	$-0.357853\pi$
$269$	14.1665	0.863747	0.431873	+	0.901934i	$0.357853\pi$
$270$	0	0
$271$	15.5830	0.946600	0.473300	−	0.880901i	$-0.343063\pi$
$271$	15.5830	0.946600	0.473300	+	0.880901i	$0.343063\pi$
$272$	0	0
$273$	−20.9830	−1.26995
$274$	0	0
$275$	3.64331	0.219700
$276$	0	0
$277$	0.412937	0.0248110	0.0124055	−	0.999923i	$-0.496051\pi$
$277$	0.412937	0.0248110	0.0124055	+	0.999923i	$0.496051\pi$
$278$	0	0
$279$	6.52881	0.390870
$280$	0	0
$281$	−18.9951	−1.13315	−0.566577	−	0.824009i	$-0.691733\pi$
$281$	−18.9951	−1.13315	−0.566577	+	0.824009i	$0.691733\pi$
$282$	0	0
$283$	−1.39753	−0.0830748	−0.0415374	−	0.999137i	$-0.513226\pi$
$283$	−1.39753	−0.0830748	−0.0415374	+	0.999137i	$0.513226\pi$
$284$	0	0
$285$	20.6546	1.22347
$286$	0	0
$287$	−0.00579487	−0.000342060 0
$288$	0	0
$289$	16.8381	0.990476
$290$	0	0
$291$	−37.4696	−2.19651
$292$	0	0
$293$	−5.10362	−0.298156	−0.149078	−	0.988825i	$-0.547631\pi$
$293$	−5.10362	−0.298156	−0.149078	+	0.988825i	$0.547631\pi$
$294$	0	0
$295$	5.57456	0.324564
$296$	0	0
$297$	0.241770	0.0140289
$298$	0	0
$299$	−2.80942	−0.162473
$300$	0	0
$301$	−7.85651	−0.452842
$302$	0	0
$303$	48.3991	2.78046
$304$	0	0
$305$	−11.4481	−0.655515
$306$	0	0
$307$	−16.4210	−0.937197	−0.468599	−	0.883411i	$-0.655241\pi$
$307$	−16.4210	−0.937197	−0.468599	+	0.883411i	$0.655241\pi$
$308$	0	0
$309$	3.15141	0.179277
$310$	0	0
$311$	−13.1430	−0.745269	−0.372635	−	0.927978i	$-0.621545\pi$
$311$	−13.1430	−0.745269	−0.372635	+	0.927978i	$0.621545\pi$
$312$	0	0
$313$	−25.8920	−1.46350	−0.731750	−	0.681573i	$-0.761296\pi$
$313$	−25.8920	−1.46350	−0.731750	+	0.681573i	$0.761296\pi$
$314$	0	0
$315$	−21.4264	−1.20724
$316$	0	0
$317$	−15.7475	−0.884468	−0.442234	−	0.896900i	$-0.645814\pi$
$317$	−15.7475	−0.884468	−0.442234	+	0.896900i	$0.645814\pi$
$318$	0	0
$319$	−8.47028	−0.474245
$320$	0	0
$321$	24.6387	1.37520
$322$	0	0
$323$	−31.5401	−1.75494
$324$	0	0
$325$	4.92365	0.273115
$326$	0	0
$327$	−31.9250	−1.76546
$328$	0	0
$329$	20.1959	1.11344
$330$	0	0
$331$	−6.21265	−0.341478	−0.170739	−	0.985316i	$-0.554615\pi$
$331$	−6.21265	−0.341478	−0.170739	+	0.985316i	$0.554615\pi$
$332$	0	0
$333$	18.1246	0.993223
$334$	0	0
$335$	16.5602	0.904781
$336$	0	0
$337$	−1.31220	−0.0714798	−0.0357399	−	0.999361i	$-0.511379\pi$
$337$	−1.31220	−0.0714798	−0.0357399	+	0.999361i	$0.511379\pi$
$338$	0	0
$339$	−16.3731	−0.889265
$340$	0	0
$341$	−2.96899	−0.160780
$342$	0	0
$343$	−28.7611	−1.55295
$344$	0	0
$345$	−5.67190	−0.305365
$346$	0	0
$347$	−31.3057	−1.68058	−0.840290	−	0.542138i	$-0.817615\pi$
$347$	−31.3057	−1.68058	−0.840290	+	0.542138i	$0.817615\pi$
$348$	0	0
$349$	4.59932	0.246196	0.123098	−	0.992395i	$-0.460717\pi$
$349$	4.59932	0.246196	0.123098	+	0.992395i	$0.460717\pi$
$350$	0	0
$351$	0.326734	0.0174398
$352$	0	0
$353$	−19.6863	−1.04779	−0.523897	−	0.851782i	$-0.675522\pi$
$353$	−19.6863	−1.04779	−0.523897	+	0.851782i	$0.675522\pi$
$354$	0	0
$355$	10.1881	0.540727
$356$	0	0
$357$	64.6883	3.42367
$358$	0	0
$359$	−23.9206	−1.26248	−0.631241	−	0.775587i	$-0.717454\pi$
$359$	−23.9206	−1.26248	−0.631241	+	0.775587i	$0.717454\pi$
$360$	0	0
$361$	10.3982	0.547276
$362$	0	0
$363$	22.2988	1.17038
$364$	0	0
$365$	−12.7428	−0.666989
$366$	0	0
$367$	−25.8401	−1.34884	−0.674421	−	0.738347i	$-0.735607\pi$
$367$	−25.8401	−1.34884	−0.674421	+	0.738347i	$0.735607\pi$
$368$	0	0
$369$	0.00394188	0.000205206 0
$370$	0	0
$371$	−5.04147	−0.261740
$372$	0	0
$373$	2.91153	0.150753	0.0753767	−	0.997155i	$-0.475984\pi$
$373$	2.91153	0.150753	0.0753767	+	0.997155i	$0.475984\pi$
$374$	0	0
$375$	28.9873	1.49690
$376$	0	0
$377$	−11.4469	−0.589547
$378$	0	0
$379$	−7.47288	−0.383856	−0.191928	−	0.981409i	$-0.561474\pi$
$379$	−7.47288	−0.383856	−0.191928	+	0.981409i	$0.561474\pi$
$380$	0	0
$381$	20.3660	1.04338
$382$	0	0
$383$	−21.1650	−1.08148	−0.540740	−	0.841190i	$-0.681856\pi$
$383$	−21.1650	−1.08148	−0.540740	+	0.841190i	$0.681856\pi$
$384$	0	0
$385$	9.74369	0.496584
$386$	0	0
$387$	5.34428	0.271665
$388$	0	0
$389$	25.8454	1.31041	0.655207	−	0.755450i	$-0.272582\pi$
$389$	25.8454	1.31041	0.655207	+	0.755450i	$0.272582\pi$
$390$	0	0
$391$	8.66114	0.438013
$392$	0	0
$393$	−50.2885	−2.53672
$394$	0	0
$395$	10.9999	0.553463
$396$	0	0
$397$	−8.17611	−0.410347	−0.205174	−	0.978726i	$-0.565776\pi$
$397$	−8.17611	−0.410347	−0.205174	+	0.978726i	$0.565776\pi$
$398$	0	0
$399$	−60.2953	−3.01854
$400$	0	0
$401$	11.8114	0.589832	0.294916	−	0.955523i	$-0.404708\pi$
$401$	11.8114	0.589832	0.294916	+	0.955523i	$0.404708\pi$
$402$	0	0
$403$	−4.01236	−0.199870
$404$	0	0
$405$	−13.5817	−0.674881
$406$	0	0
$407$	−8.24221	−0.408551
$408$	0	0
$409$	19.2109	0.949918	0.474959	−	0.880008i	$-0.342463\pi$
$409$	19.2109	0.949918	0.474959	+	0.880008i	$0.342463\pi$
$410$	0	0
$411$	31.6732	1.56232
$412$	0	0
$413$	−16.2734	−0.800760
$414$	0	0
$415$	5.45566	0.267808
$416$	0	0
$417$	−23.3832	−1.14508
$418$	0	0
$419$	−18.6162	−0.909459	−0.454730	−	0.890629i	$-0.650264\pi$
$419$	−18.6162	−0.909459	−0.454730	+	0.890629i	$0.650264\pi$
$420$	0	0
$421$	−34.4681	−1.67987	−0.839937	−	0.542684i	$-0.817408\pi$
$421$	−34.4681	−1.67987	−0.839937	+	0.542684i	$0.817408\pi$
$422$	0	0
$423$	−13.7380	−0.667965
$424$	0	0
$425$	−15.1791	−0.736295
$426$	0	0
$427$	33.4194	1.61728
$428$	0	0
$429$	−6.49085	−0.313381
$430$	0	0
$431$	20.1809	0.972081	0.486040	−	0.873936i	$-0.338441\pi$
$431$	20.1809	0.972081	0.486040	+	0.873936i	$0.338441\pi$
$432$	0	0
$433$	−33.5443	−1.61204	−0.806019	−	0.591889i	$-0.798382\pi$
$433$	−33.5443	−1.61204	−0.806019	+	0.591889i	$0.798382\pi$
$434$	0	0
$435$	−23.1101	−1.10804
$436$	0	0
$437$	−8.07296	−0.386182
$438$	0	0
$439$	4.27741	0.204150	0.102075	−	0.994777i	$-0.467452\pi$
$439$	4.27741	0.204150	0.102075	+	0.994777i	$0.467452\pi$
$440$	0	0
$441$	41.0563	1.95506
$442$	0	0
$443$	−26.1991	−1.24476	−0.622378	−	0.782716i	$-0.713833\pi$
$443$	−26.1991	−1.24476	−0.622378	+	0.782716i	$0.713833\pi$
$444$	0	0
$445$	−16.4397	−0.779315
$446$	0	0
$447$	2.31948	0.109708
$448$	0	0
$449$	−10.8570	−0.512373	−0.256186	−	0.966627i	$-0.582466\pi$
$449$	−10.8570	−0.512373	−0.256186	+	0.966627i	$0.582466\pi$
$450$	0	0
$451$	−0.00179258	−8.44092e−5 0
$452$	0	0
$453$	−1.81783	−0.0854092
$454$	0	0
$455$	13.1678	0.617318
$456$	0	0
$457$	−34.0541	−1.59298	−0.796492	−	0.604649i	$-0.793313\pi$
$457$	−34.0541	−1.59298	−0.796492	+	0.604649i	$0.793313\pi$
$458$	0	0
$459$	−1.00729	−0.0470161
$460$	0	0
$461$	−21.7221	−1.01170	−0.505851	−	0.862621i	$-0.668821\pi$
$461$	−21.7221	−1.01170	−0.505851	+	0.862621i	$0.668821\pi$
$462$	0	0
$463$	−18.8802	−0.877437	−0.438719	−	0.898624i	$-0.644568\pi$
$463$	−18.8802	−0.877437	−0.438719	+	0.898624i	$0.644568\pi$
$464$	0	0
$465$	−8.10050	−0.375652
$466$	0	0
$467$	9.44774	0.437189	0.218595	−	0.975816i	$-0.429853\pi$
$467$	9.44774	0.437189	0.218595	+	0.975816i	$0.429853\pi$
$468$	0	0
$469$	−48.3429	−2.23227
$470$	0	0
$471$	21.9819	1.01287
$472$	0	0
$473$	−2.43032	−0.111746
$474$	0	0
$475$	14.1483	0.649168
$476$	0	0
$477$	3.42939	0.157021
$478$	0	0
$479$	3.66074	0.167263	0.0836317	−	0.996497i	$-0.473348\pi$
$479$	3.66074	0.167263	0.0836317	+	0.996497i	$0.473348\pi$
$480$	0	0
$481$	−11.1387	−0.507881
$482$	0	0
$483$	16.5575	0.753393
$484$	0	0
$485$	23.5140	1.06772
$486$	0	0
$487$	−9.16552	−0.415329	−0.207665	−	0.978200i	$-0.566586\pi$
$487$	−9.16552	−0.415329	−0.207665	+	0.978200i	$0.566586\pi$
$488$	0	0
$489$	49.1922	2.22455
$490$	0	0
$491$	24.2323	1.09359	0.546794	−	0.837267i	$-0.315848\pi$
$491$	24.2323	1.09359	0.546794	+	0.837267i	$0.315848\pi$
$492$	0	0
$493$	35.2897	1.58937
$494$	0	0
$495$	−6.62801	−0.297907
$496$	0	0
$497$	−29.7412	−1.33408
$498$	0	0
$499$	−0.463648	−0.0207557	−0.0103779	−	0.999946i	$-0.503303\pi$
$499$	−0.463648	−0.0207557	−0.0103779	+	0.999946i	$0.503303\pi$
$500$	0	0
$501$	38.5203	1.72096
$502$	0	0
$503$	−18.5312	−0.826265	−0.413133	−	0.910671i	$-0.635565\pi$
$503$	−18.5312	−0.826265	−0.413133	+	0.910671i	$0.635565\pi$
$504$	0	0
$505$	−30.3728	−1.35157
$506$	0	0
$507$	23.2575	1.03290
$508$	0	0
$509$	−1.92753	−0.0854362	−0.0427181	−	0.999087i	$-0.513602\pi$
$509$	−1.92753	−0.0854362	−0.0427181	+	0.999087i	$0.513602\pi$
$510$	0	0
$511$	37.1991	1.64559
$512$	0	0
$513$	0.938882	0.0414527
$514$	0	0
$515$	−1.97766	−0.0871462
$516$	0	0
$517$	6.24738	0.274760
$518$	0	0
$519$	−0.654483	−0.0287286
$520$	0	0
$521$	17.3717	0.761066	0.380533	−	0.924767i	$-0.375741\pi$
$521$	17.3717	0.761066	0.380533	+	0.924767i	$0.375741\pi$
$522$	0	0
$523$	23.3181	1.01963	0.509815	−	0.860284i	$-0.329714\pi$
$523$	23.3181	1.01963	0.509815	+	0.860284i	$0.329714\pi$
$524$	0	0
$525$	−29.0179	−1.26645
$526$	0	0
$527$	12.3697	0.538832
$528$	0	0
$529$	−20.7831	−0.903613
$530$	0	0
$531$	11.0697	0.480386
$532$	0	0
$533$	−0.00242253	−0.000104931 0
$534$	0	0
$535$	−15.4620	−0.668479
$536$	0	0
$537$	−26.5479	−1.14563
$538$	0	0
$539$	−18.6704	−0.804193
$540$	0	0
$541$	16.6818	0.717207	0.358603	−	0.933490i	$-0.383253\pi$
$541$	16.6818	0.717207	0.358603	+	0.933490i	$0.383253\pi$
$542$	0	0
$543$	−25.1687	−1.08009
$544$	0	0
$545$	20.0345	0.858184
$546$	0	0
$547$	19.8737	0.849740	0.424870	−	0.905254i	$-0.360320\pi$
$547$	19.8737	0.849740	0.424870	+	0.905254i	$0.360320\pi$
$548$	0	0
$549$	−22.7331	−0.970226
$550$	0	0
$551$	−32.8932	−1.40130
$552$	0	0
$553$	−32.1110	−1.36550
$554$	0	0
$555$	−22.4878	−0.954554
$556$	0	0
$557$	37.2954	1.58026	0.790128	−	0.612941i	$-0.210014\pi$
$557$	37.2954	1.58026	0.790128	+	0.612941i	$0.210014\pi$
$558$	0	0
$559$	−3.28439	−0.138915
$560$	0	0
$561$	20.0106	0.844848
$562$	0	0
$563$	18.0505	0.760737	0.380369	−	0.924835i	$-0.375797\pi$
$563$	18.0505	0.760737	0.380369	+	0.924835i	$0.375797\pi$
$564$	0	0
$565$	10.2749	0.432269
$566$	0	0
$567$	39.6480	1.66506
$568$	0	0
$569$	−12.6665	−0.531007	−0.265503	−	0.964110i	$-0.585538\pi$
$569$	−12.6665	−0.531007	−0.265503	+	0.964110i	$0.585538\pi$
$570$	0	0
$571$	35.1787	1.47218	0.736092	−	0.676882i	$-0.236669\pi$
$571$	35.1787	1.47218	0.736092	+	0.676882i	$0.236669\pi$
$572$	0	0
$573$	−19.2236	−0.803079
$574$	0	0
$575$	−3.88522	−0.162025
$576$	0	0
$577$	10.3487	0.430823	0.215411	−	0.976523i	$-0.430891\pi$
$577$	10.3487	0.430823	0.215411	+	0.976523i	$0.430891\pi$
$578$	0	0
$579$	−55.5355	−2.30798
$580$	0	0
$581$	−15.9263	−0.660732
$582$	0	0
$583$	−1.55952	−0.0645888
$584$	0	0
$585$	−8.95724	−0.370336
$586$	0	0
$587$	−33.9217	−1.40010	−0.700050	−	0.714094i	$-0.746839\pi$
$587$	−33.9217	−1.40010	−0.700050	+	0.714094i	$0.746839\pi$
$588$	0	0
$589$	−11.5297	−0.475071
$590$	0	0
$591$	14.7972	0.608677
$592$	0	0
$593$	−32.9176	−1.35176	−0.675882	−	0.737010i	$-0.736237\pi$
$593$	−32.9176	−1.35176	−0.675882	+	0.737010i	$0.736237\pi$
$594$	0	0
$595$	−40.5951	−1.66424
$596$	0	0
$597$	−24.3601	−0.996993
$598$	0	0
$599$	3.31308	0.135369	0.0676844	−	0.997707i	$-0.478439\pi$
$599$	3.31308	0.135369	0.0676844	+	0.997707i	$0.478439\pi$
$600$	0	0
$601$	20.3412	0.829734	0.414867	−	0.909882i	$-0.363828\pi$
$601$	20.3412	0.829734	0.414867	+	0.909882i	$0.363828\pi$
$602$	0	0
$603$	32.8846	1.33916
$604$	0	0
$605$	−13.9936	−0.568919
$606$	0	0
$607$	−12.5095	−0.507747	−0.253873	−	0.967237i	$-0.581705\pi$
$607$	−12.5095	−0.507747	−0.253873	+	0.967237i	$0.581705\pi$
$608$	0	0
$609$	67.4633	2.73375
$610$	0	0
$611$	8.44285	0.341561
$612$	0	0
$613$	39.6818	1.60273	0.801367	−	0.598173i	$-0.204107\pi$
$613$	39.6818	1.60273	0.801367	+	0.598173i	$0.204107\pi$
$614$	0	0
$615$	−0.00489082	−0.000197217 0
$616$	0	0
$617$	−21.4096	−0.861919	−0.430959	−	0.902371i	$-0.641825\pi$
$617$	−21.4096	−0.861919	−0.430959	+	0.902371i	$0.641825\pi$
$618$	0	0
$619$	−12.5691	−0.505195	−0.252597	−	0.967571i	$-0.581285\pi$
$619$	−12.5691	−0.505195	−0.252597	+	0.967571i	$0.581285\pi$
$620$	0	0
$621$	−0.257823	−0.0103461
$622$	0	0
$623$	47.9910	1.92272
$624$	0	0
$625$	−5.14386	−0.205754
$626$	0	0
$627$	−18.6517	−0.744877
$628$	0	0
$629$	34.3395	1.36920
$630$	0	0
$631$	−42.1760	−1.67900	−0.839499	−	0.543361i	$-0.817152\pi$
$631$	−42.1760	−1.67900	−0.839499	+	0.543361i	$0.817152\pi$
$632$	0	0
$633$	−12.4268	−0.493921
$634$	0	0
$635$	−12.7807	−0.507186
$636$	0	0
$637$	−25.2316	−0.999714
$638$	0	0
$639$	20.2311	0.800328
$640$	0	0
$641$	−4.85892	−0.191916	−0.0959579	−	0.995385i	$-0.530591\pi$
$641$	−4.85892	−0.191916	−0.0959579	+	0.995385i	$0.530591\pi$
$642$	0	0
$643$	−23.8815	−0.941795	−0.470897	−	0.882188i	$-0.656070\pi$
$643$	−23.8815	−0.941795	−0.470897	+	0.882188i	$0.656070\pi$
$644$	0	0
$645$	−6.63082	−0.261088
$646$	0	0
$647$	−3.71552	−0.146072	−0.0730361	−	0.997329i	$-0.523269\pi$
$647$	−3.71552	−0.146072	−0.0730361	+	0.997329i	$0.523269\pi$
$648$	0	0
$649$	−5.03398	−0.197601
$650$	0	0
$651$	23.6471	0.926805
$652$	0	0
$653$	23.7542	0.929574	0.464787	−	0.885422i	$-0.346131\pi$
$653$	23.7542	0.929574	0.464787	+	0.885422i	$0.346131\pi$
$654$	0	0
$655$	31.5585	1.23309
$656$	0	0
$657$	−25.3041	−0.987209
$658$	0	0
$659$	23.0706	0.898704	0.449352	−	0.893355i	$-0.351655\pi$
$659$	23.0706	0.898704	0.449352	+	0.893355i	$0.351655\pi$
$660$	0	0
$661$	0.154231	0.00599891	0.00299945	−	0.999996i	$-0.499045\pi$
$661$	0.154231	0.00599891	0.00299945	+	0.999996i	$0.499045\pi$
$662$	0	0
$663$	27.0428	1.05025
$664$	0	0
$665$	37.8383	1.46730
$666$	0	0
$667$	9.03269	0.349747
$668$	0	0
$669$	26.5328	1.02582
$670$	0	0
$671$	10.3379	0.399091
$672$	0	0
$673$	−25.5536	−0.985021	−0.492510	−	0.870307i	$-0.663921\pi$
$673$	−25.5536	−0.985021	−0.492510	+	0.870307i	$0.663921\pi$
$674$	0	0
$675$	0.451849	0.0173917
$676$	0	0
$677$	34.1875	1.31393	0.656965	−	0.753921i	$-0.271840\pi$
$677$	34.1875	1.31393	0.656965	+	0.753921i	$0.271840\pi$
$678$	0	0
$679$	−68.6426	−2.63426
$680$	0	0
$681$	70.1013	2.68629
$682$	0	0
$683$	−26.7749	−1.02451	−0.512257	−	0.858832i	$-0.671190\pi$
$683$	−26.7749	−1.02451	−0.512257	+	0.858832i	$0.671190\pi$
$684$	0	0
$685$	−19.8765	−0.759442
$686$	0	0
$687$	16.1275	0.615303
$688$	0	0
$689$	−2.10757	−0.0802921
$690$	0	0
$691$	15.5457	0.591387	0.295693	−	0.955283i	$-0.404449\pi$
$691$	15.5457	0.591387	0.295693	+	0.955283i	$0.404449\pi$
$692$	0	0
$693$	19.3486	0.734993
$694$	0	0
$695$	14.6741	0.556620
$696$	0	0
$697$	0.00746841	0.000282886 0
$698$	0	0
$699$	33.5880	1.27042
$700$	0	0
$701$	−5.54151	−0.209300	−0.104650	−	0.994509i	$-0.533372\pi$
$701$	−5.54151	−0.209300	−0.104650	+	0.994509i	$0.533372\pi$
$702$	0	0
$703$	−32.0075	−1.20718
$704$	0	0
$705$	17.0452	0.641959
$706$	0	0
$707$	88.6649	3.33458
$708$	0	0
$709$	−0.343763	−0.0129103	−0.00645514	−	0.999979i	$-0.502055\pi$
$709$	−0.343763	−0.0129103	−0.00645514	+	0.999979i	$0.502055\pi$
$710$	0	0
$711$	21.8431	0.819179
$712$	0	0
$713$	3.16612	0.118572
$714$	0	0
$715$	4.07332	0.152334
$716$	0	0
$717$	−21.3891	−0.798792
$718$	0	0
$719$	−33.2707	−1.24079	−0.620394	−	0.784290i	$-0.713027\pi$
$719$	−33.2707	−1.24079	−0.620394	+	0.784290i	$0.713027\pi$
$720$	0	0
$721$	5.77323	0.215006
$722$	0	0
$723$	−14.9812	−0.557158
$724$	0	0
$725$	−15.8303	−0.587921
$726$	0	0
$727$	−50.5555	−1.87500	−0.937499	−	0.347988i	$-0.886865\pi$
$727$	−50.5555	−1.87500	−0.937499	+	0.347988i	$0.886865\pi$
$728$	0	0
$729$	−28.2499	−1.04629
$730$	0	0
$731$	10.1254	0.374503
$732$	0	0
$733$	−39.1140	−1.44471	−0.722354	−	0.691523i	$-0.756940\pi$
$733$	−39.1140	−1.44471	−0.722354	+	0.691523i	$0.756940\pi$
$734$	0	0
$735$	−50.9399	−1.87895
$736$	0	0
$737$	−14.9543	−0.550850
$738$	0	0
$739$	42.9975	1.58169	0.790844	−	0.612018i	$-0.209642\pi$
$739$	42.9975	1.58169	0.790844	+	0.612018i	$0.209642\pi$
$740$	0	0
$741$	−25.2063	−0.925977
$742$	0	0
$743$	48.7392	1.78807	0.894035	−	0.447998i	$-0.147863\pi$
$743$	48.7392	1.78807	0.894035	+	0.447998i	$0.147863\pi$
$744$	0	0
$745$	−1.45559	−0.0533286
$746$	0	0
$747$	10.8336	0.396381
$748$	0	0
$749$	45.1368	1.64926
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−4.29905	−0.156666
$754$	0	0
$755$	1.14078	0.0415172
$756$	0	0
$757$	−31.0828	−1.12973	−0.564863	−	0.825185i	$-0.691071\pi$
$757$	−31.0828	−1.12973	−0.564863	+	0.825185i	$0.691071\pi$
$758$	0	0
$759$	5.12188	0.185913
$760$	0	0
$761$	−5.55942	−0.201529	−0.100764	−	0.994910i	$-0.532129\pi$
$761$	−5.55942	−0.201529	−0.100764	+	0.994910i	$0.532129\pi$
$762$	0	0
$763$	−58.4851	−2.11730
$764$	0	0
$765$	27.6142	0.998395
$766$	0	0
$767$	−6.80304	−0.245643
$768$	0	0
$769$	13.8881	0.500816	0.250408	−	0.968140i	$-0.419435\pi$
$769$	13.8881	0.500816	0.250408	+	0.968140i	$0.419435\pi$
$770$	0	0
$771$	−29.2485	−1.05336
$772$	0	0
$773$	25.7487	0.926117	0.463059	−	0.886328i	$-0.346752\pi$
$773$	25.7487	0.926117	0.463059	+	0.886328i	$0.346752\pi$
$774$	0	0
$775$	−5.54880	−0.199319
$776$	0	0
$777$	65.6468	2.35507
$778$	0	0
$779$	−0.00696123	−0.000249412 0
$780$	0	0
$781$	−9.20012	−0.329206
$782$	0	0
$783$	−1.05050	−0.0375417
$784$	0	0
$785$	−13.7947	−0.492354
$786$	0	0
$787$	27.9233	0.995358	0.497679	−	0.867361i	$-0.334186\pi$
$787$	27.9233	0.995358	0.497679	+	0.867361i	$0.334186\pi$
$788$	0	0
$789$	−27.2751	−0.971019
$790$	0	0
$791$	−29.9947	−1.06649
$792$	0	0
$793$	13.9709	0.496121
$794$	0	0
$795$	−4.25495	−0.150908
$796$	0	0
$797$	52.5312	1.86075	0.930375	−	0.366609i	$-0.119481\pi$
$797$	52.5312	1.86075	0.930375	+	0.366609i	$0.119481\pi$
$798$	0	0
$799$	−26.0284	−0.920820
$800$	0	0
$801$	−32.6452	−1.15346
$802$	0	0
$803$	11.5071	0.406077
$804$	0	0
$805$	−10.3906	−0.366222
$806$	0	0
$807$	−34.9033	−1.22866
$808$	0	0
$809$	−32.3488	−1.13732	−0.568661	−	0.822572i	$-0.692539\pi$
$809$	−32.3488	−1.13732	−0.568661	+	0.822572i	$0.692539\pi$
$810$	0	0
$811$	32.4210	1.13846	0.569229	−	0.822179i	$-0.307242\pi$
$811$	32.4210	1.13846	0.569229	+	0.822179i	$0.307242\pi$
$812$	0	0
$813$	−38.3933	−1.34651
$814$	0	0
$815$	−30.8705	−1.08135
$816$	0	0
$817$	−9.43782	−0.330188
$818$	0	0
$819$	26.1481	0.913690
$820$	0	0
$821$	32.4244	1.13162	0.565810	−	0.824536i	$-0.308563\pi$
$821$	32.4244	1.13162	0.565810	+	0.824536i	$0.308563\pi$
$822$	0	0
$823$	−45.2888	−1.57867	−0.789335	−	0.613963i	$-0.789574\pi$
$823$	−45.2888	−1.57867	−0.789335	+	0.613963i	$0.789574\pi$
$824$	0	0
$825$	−8.97637	−0.312517
$826$	0	0
$827$	6.20933	0.215919	0.107960	−	0.994155i	$-0.465568\pi$
$827$	6.20933	0.215919	0.107960	+	0.994155i	$0.465568\pi$
$828$	0	0
$829$	−26.6197	−0.924542	−0.462271	−	0.886739i	$-0.652965\pi$
$829$	−26.6197	−0.924542	−0.462271	+	0.886739i	$0.652965\pi$
$830$	0	0
$831$	−1.01739	−0.0352930
$832$	0	0
$833$	77.7865	2.69514
$834$	0	0
$835$	−24.1734	−0.836553
$836$	0	0
$837$	−0.368219	−0.0127275
$838$	0	0
$839$	−38.1136	−1.31583	−0.657914	−	0.753093i	$-0.728561\pi$
$839$	−38.1136	−1.31583	−0.657914	+	0.753093i	$0.728561\pi$
$840$	0	0
$841$	7.80355	0.269088
$842$	0	0
$843$	46.8000	1.61188
$844$	0	0
$845$	−14.5952	−0.502090
$846$	0	0
$847$	40.8503	1.40363
$848$	0	0
$849$	3.44324	0.118172
$850$	0	0
$851$	8.78947	0.301299
$852$	0	0
$853$	41.0705	1.40623	0.703113	−	0.711078i	$-0.251793\pi$
$853$	41.0705	1.40623	0.703113	+	0.711078i	$0.251793\pi$
$854$	0	0
$855$	−25.7390	−0.880254
$856$	0	0
$857$	9.40547	0.321285	0.160642	−	0.987013i	$-0.448643\pi$
$857$	9.40547	0.321285	0.160642	+	0.987013i	$0.448643\pi$
$858$	0	0
$859$	8.74773	0.298469	0.149234	−	0.988802i	$-0.452319\pi$
$859$	8.74773	0.298469	0.149234	+	0.988802i	$0.452319\pi$
$860$	0	0
$861$	0.0142774	0.000486571 0
$862$	0	0
$863$	−29.6683	−1.00992	−0.504960	−	0.863143i	$-0.668493\pi$
$863$	−29.6683	−1.00992	−0.504960	+	0.863143i	$0.668493\pi$
$864$	0	0
$865$	0.410720	0.0139649
$866$	0	0
$867$	−41.4856	−1.40892
$868$	0	0
$869$	−9.93317	−0.336960
$870$	0	0
$871$	−20.2096	−0.684776
$872$	0	0
$873$	46.6932	1.58032
$874$	0	0
$875$	53.1033	1.79522
$876$	0	0
$877$	−50.7500	−1.71370	−0.856852	−	0.515562i	$-0.827583\pi$
$877$	−50.7500	−1.71370	−0.856852	+	0.515562i	$0.827583\pi$
$878$	0	0
$879$	12.5743	0.424119
$880$	0	0
$881$	−9.19753	−0.309873	−0.154936	−	0.987924i	$-0.549517\pi$
$881$	−9.19753	−0.309873	−0.154936	+	0.987924i	$0.549517\pi$
$882$	0	0
$883$	25.8988	0.871564	0.435782	−	0.900052i	$-0.356472\pi$
$883$	25.8988	0.871564	0.435782	+	0.900052i	$0.356472\pi$
$884$	0	0
$885$	−13.7346	−0.461683
$886$	0	0
$887$	40.0363	1.34429	0.672144	−	0.740420i	$-0.265374\pi$
$887$	40.0363	1.34429	0.672144	+	0.740420i	$0.265374\pi$
$888$	0	0
$889$	37.3096	1.25132
$890$	0	0
$891$	12.2647	0.410882
$892$	0	0
$893$	24.2608	0.811858
$894$	0	0
$895$	16.6601	0.556886
$896$	0	0
$897$	6.92182	0.231113
$898$	0	0
$899$	12.9003	0.430250
$900$	0	0
$901$	6.49742	0.216461
$902$	0	0
$903$	19.3568	0.644155
$904$	0	0
$905$	15.7946	0.525030
$906$	0	0
$907$	−16.3952	−0.544393	−0.272197	−	0.962242i	$-0.587750\pi$
$907$	−16.3952	−0.544393	−0.272197	+	0.962242i	$0.587750\pi$
$908$	0	0
$909$	−60.3130	−2.00046
$910$	0	0
$911$	−45.3056	−1.50104	−0.750521	−	0.660846i	$-0.770198\pi$
$911$	−45.3056	−1.50104	−0.750521	+	0.660846i	$0.770198\pi$
$912$	0	0
$913$	−4.92661	−0.163047
$914$	0	0
$915$	28.2057	0.932452
$916$	0	0
$917$	−92.1262	−3.04227
$918$	0	0
$919$	13.2689	0.437699	0.218850	−	0.975759i	$-0.429770\pi$
$919$	13.2689	0.437699	0.218850	+	0.975759i	$0.429770\pi$
$920$	0	0
$921$	40.4580	1.33314
$922$	0	0
$923$	−12.4332	−0.409245
$924$	0	0
$925$	−15.4040	−0.506481
$926$	0	0
$927$	−3.92716	−0.128985
$928$	0	0
$929$	26.3443	0.864328	0.432164	−	0.901795i	$-0.357750\pi$
$929$	26.3443	0.864328	0.432164	+	0.901795i	$0.357750\pi$
$930$	0	0
$931$	−72.5041	−2.37622
$932$	0	0
$933$	32.3816	1.06012
$934$	0	0
$935$	−12.5576	−0.410678
$936$	0	0
$937$	1.65737	0.0541440	0.0270720	−	0.999633i	$-0.491382\pi$
$937$	1.65737	0.0541440	0.0270720	+	0.999633i	$0.491382\pi$
$938$	0	0
$939$	63.7924	2.08179
$940$	0	0
$941$	−7.52502	−0.245309	−0.122654	−	0.992449i	$-0.539141\pi$
$941$	−7.52502	−0.245309	−0.122654	+	0.992449i	$0.539141\pi$
$942$	0	0
$943$	0.00191160	6.22503e−5 0
$944$	0	0
$945$	1.20843	0.0393102
$946$	0	0
$947$	−51.6190	−1.67739	−0.838696	−	0.544599i	$-0.816682\pi$
$947$	−51.6190	−1.67739	−0.838696	+	0.544599i	$0.816682\pi$
$948$	0	0
$949$	15.5510	0.504806
$950$	0	0
$951$	38.7986	1.25813
$952$	0	0
$953$	13.2680	0.429793	0.214897	−	0.976637i	$-0.431059\pi$
$953$	13.2680	0.429793	0.214897	+	0.976637i	$0.431059\pi$
$954$	0	0
$955$	12.0638	0.390374
$956$	0	0
$957$	20.8690	0.674600
$958$	0	0
$959$	58.0238	1.87369
$960$	0	0
$961$	−26.4782	−0.854135
$962$	0	0
$963$	−30.7037	−0.989413
$964$	0	0
$965$	34.8512	1.12190
$966$	0	0
$967$	9.82786	0.316043	0.158021	−	0.987436i	$-0.449489\pi$
$967$	9.82786	0.316043	0.158021	+	0.987436i	$0.449489\pi$
$968$	0	0
$969$	77.7084	2.49635
$970$	0	0
$971$	−41.3998	−1.32858	−0.664292	−	0.747473i	$-0.731267\pi$
$971$	−41.3998	−1.32858	−0.664292	+	0.747473i	$0.731267\pi$
$972$	0	0
$973$	−42.8369	−1.37329
$974$	0	0
$975$	−12.1309	−0.388499
$976$	0	0
$977$	−12.3075	−0.393751	−0.196876	−	0.980428i	$-0.563080\pi$
$977$	−12.3075	−0.393751	−0.196876	+	0.980428i	$0.563080\pi$
$978$	0	0
$979$	14.8455	0.474464
$980$	0	0
$981$	39.7837	1.27020
$982$	0	0
$983$	57.4653	1.83286	0.916429	−	0.400198i	$-0.131059\pi$
$983$	57.4653	1.83286	0.916429	+	0.400198i	$0.131059\pi$
$984$	0	0
$985$	−9.28598	−0.295876
$986$	0	0
$987$	−49.7586	−1.58383
$988$	0	0
$989$	2.59169	0.0824110
$990$	0	0
$991$	58.8952	1.87087	0.935433	−	0.353503i	$-0.115010\pi$
$991$	58.8952	1.87087	0.935433	+	0.353503i	$0.115010\pi$
$992$	0	0
$993$	15.3067	0.485743
$994$	0	0
$995$	15.2872	0.484636
$996$	0	0
$997$	−33.5055	−1.06113	−0.530566	−	0.847644i	$-0.678020\pi$
$997$	−33.5055	−1.06113	−0.530566	+	0.847644i	$0.678020\pi$
$998$	0	0
$999$	−1.02221	−0.0323414

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.7	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-2.46379 q^{3} +1.54615 q^{5} -4.51355 q^{7} +3.07028 q^{9} +O(q^{10})\)	\(q-2.46379 q^{3} +1.54615 q^{5} -4.51355 q^{7} +3.07028 q^{9} -1.39622 q^{11} -1.88688 q^{13} -3.80940 q^{15} +5.81705 q^{17} -5.42202 q^{19} +11.1205 q^{21} +1.48892 q^{23} -2.60942 q^{25} -0.173161 q^{27} +6.06659 q^{29} +2.12645 q^{31} +3.43999 q^{33} -6.97863 q^{35} +5.90324 q^{37} +4.64888 q^{39} +0.00128388 q^{41} +1.74065 q^{43} +4.74712 q^{45} -4.47451 q^{47} +13.3722 q^{49} -14.3320 q^{51} +1.11696 q^{53} -2.15876 q^{55} +13.3587 q^{57} +3.60544 q^{59} -7.40424 q^{61} -13.8579 q^{63} -2.91740 q^{65} +10.7106 q^{67} -3.66840 q^{69} +6.58932 q^{71} -8.24164 q^{73} +6.42907 q^{75} +6.30190 q^{77} +7.11435 q^{79} -8.78421 q^{81} +3.52854 q^{83} +8.99404 q^{85} -14.9468 q^{87} -10.6326 q^{89} +8.51653 q^{91} -5.23914 q^{93} -8.38325 q^{95} +15.2081 q^{97} -4.28678 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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