LMFDB - Embedded newform 6008.2.a.d.1.15

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:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-1.11974 q^{3} +2.58610 q^{5} +0.902449 q^{7} -1.74619 q^{9} +O(q^{10})$	$q-1.11974 q^{3} +2.58610 q^{5} +0.902449 q^{7} -1.74619 q^{9} +4.37607 q^{11} -0.575054 q^{13} -2.89575 q^{15} -0.596745 q^{17} -2.17299 q^{19} -1.01051 q^{21} -2.92818 q^{23} +1.68789 q^{25} +5.31449 q^{27} +7.59391 q^{29} -6.25763 q^{31} -4.90006 q^{33} +2.33382 q^{35} -1.33073 q^{37} +0.643910 q^{39} +1.89065 q^{41} +12.0288 q^{43} -4.51580 q^{45} +3.13660 q^{47} -6.18559 q^{49} +0.668199 q^{51} +7.54171 q^{53} +11.3169 q^{55} +2.43318 q^{57} +3.10098 q^{59} -6.45188 q^{61} -1.57584 q^{63} -1.48715 q^{65} +14.2787 q^{67} +3.27880 q^{69} -0.0385659 q^{71} -8.42926 q^{73} -1.89000 q^{75} +3.94918 q^{77} +6.34507 q^{79} -0.712282 q^{81} -0.391392 q^{83} -1.54324 q^{85} -8.50319 q^{87} +16.9059 q^{89} -0.518957 q^{91} +7.00691 q^{93} -5.61957 q^{95} -10.8771 q^{97} -7.64143 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * 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
$3$	−1.11974	−0.646481	−0.323241	−	0.946317i	$-0.604772\pi$
$3$	−1.11974	−0.646481	−0.323241	+	0.946317i	$0.604772\pi$
$4$	0	0
$5$	2.58610	1.15654	0.578269	−	0.815846i	$-0.303729\pi$
$5$	2.58610	1.15654	0.578269	+	0.815846i	$0.303729\pi$
$6$	0	0
$7$	0.902449	0.341093	0.170547	−	0.985350i	$-0.445447\pi$
$7$	0.902449	0.341093	0.170547	+	0.985350i	$0.445447\pi$
$8$	0	0
$9$	−1.74619	−0.582062
$10$	0	0
$11$	4.37607	1.31944	0.659718	−	0.751513i	$-0.270676\pi$
$11$	4.37607	1.31944	0.659718	+	0.751513i	$0.270676\pi$
$12$	0	0
$13$	−0.575054	−0.159491	−0.0797457	−	0.996815i	$-0.525411\pi$
$13$	−0.575054	−0.159491	−0.0797457	+	0.996815i	$0.525411\pi$
$14$	0	0
$15$	−2.89575	−0.747680
$16$	0	0
$17$	−0.596745	−0.144732	−0.0723660	−	0.997378i	$-0.523055\pi$
$17$	−0.596745	−0.144732	−0.0723660	+	0.997378i	$0.523055\pi$
$18$	0	0
$19$	−2.17299	−0.498519	−0.249259	−	0.968437i	$-0.580187\pi$
$19$	−2.17299	−0.498519	−0.249259	+	0.968437i	$0.580187\pi$
$20$	0	0
$21$	−1.01051	−0.220511
$22$	0	0
$23$	−2.92818	−0.610568	−0.305284	−	0.952261i	$-0.598751\pi$
$23$	−2.92818	−0.610568	−0.305284	+	0.952261i	$0.598751\pi$
$24$	0	0
$25$	1.68789	0.337579
$26$	0	0
$27$	5.31449	1.02277
$28$	0	0
$29$	7.59391	1.41015	0.705076	−	0.709131i	$-0.250913\pi$
$29$	7.59391	1.41015	0.705076	+	0.709131i	$0.250913\pi$
$30$	0	0
$31$	−6.25763	−1.12390	−0.561952	−	0.827170i	$-0.689949\pi$
$31$	−6.25763	−1.12390	−0.561952	+	0.827170i	$0.689949\pi$
$32$	0	0
$33$	−4.90006	−0.852991
$34$	0	0
$35$	2.33382	0.394487
$36$	0	0
$37$	−1.33073	−0.218770	−0.109385	−	0.993999i	$-0.534888\pi$
$37$	−1.33073	−0.218770	−0.109385	+	0.993999i	$0.534888\pi$
$38$	0	0
$39$	0.643910	0.103108
$40$	0	0
$41$	1.89065	0.295270	0.147635	−	0.989042i	$-0.452834\pi$
$41$	1.89065	0.295270	0.147635	+	0.989042i	$0.452834\pi$
$42$	0	0
$43$	12.0288	1.83438	0.917191	−	0.398449i	$-0.130451\pi$
$43$	12.0288	1.83438	0.917191	+	0.398449i	$0.130451\pi$
$44$	0	0
$45$	−4.51580	−0.673176
$46$	0	0
$47$	3.13660	0.457520	0.228760	−	0.973483i	$-0.426533\pi$
$47$	3.13660	0.457520	0.228760	+	0.973483i	$0.426533\pi$
$48$	0	0
$49$	−6.18559	−0.883655
$50$	0	0
$51$	0.668199	0.0935665
$52$	0	0
$53$	7.54171	1.03593	0.517967	−	0.855401i	$-0.326689\pi$
$53$	7.54171	1.03593	0.517967	+	0.855401i	$0.326689\pi$
$54$	0	0
$55$	11.3169	1.52598
$56$	0	0
$57$	2.43318	0.322283
$58$	0	0
$59$	3.10098	0.403714	0.201857	−	0.979415i	$-0.435302\pi$
$59$	3.10098	0.403714	0.201857	+	0.979415i	$0.435302\pi$
$60$	0	0
$61$	−6.45188	−0.826079	−0.413039	−	0.910713i	$-0.635533\pi$
$61$	−6.45188	−0.826079	−0.413039	+	0.910713i	$0.635533\pi$
$62$	0	0
$63$	−1.57584	−0.198537
$64$	0	0
$65$	−1.48715	−0.184458
$66$	0	0
$67$	14.2787	1.74442	0.872209	−	0.489134i	$-0.162687\pi$
$67$	14.2787	1.74442	0.872209	+	0.489134i	$0.162687\pi$
$68$	0	0
$69$	3.27880	0.394721
$70$	0	0
$71$	−0.0385659	−0.00457693	−0.00228846	−	0.999997i	$-0.500728\pi$
$71$	−0.0385659	−0.00457693	−0.00228846	+	0.999997i	$0.500728\pi$
$72$	0	0
$73$	−8.42926	−0.986571	−0.493285	−	0.869868i	$-0.664204\pi$
$73$	−8.42926	−0.986571	−0.493285	+	0.869868i	$0.664204\pi$
$74$	0	0
$75$	−1.89000	−0.218238
$76$	0	0
$77$	3.94918	0.450051
$78$	0	0
$79$	6.34507	0.713876	0.356938	−	0.934128i	$-0.383821\pi$
$79$	6.34507	0.713876	0.356938	+	0.934128i	$0.383821\pi$
$80$	0	0
$81$	−0.712282	−0.0791425
$82$	0	0
$83$	−0.391392	−0.0429609	−0.0214804	−	0.999769i	$-0.506838\pi$
$83$	−0.391392	−0.0429609	−0.0214804	+	0.999769i	$0.506838\pi$
$84$	0	0
$85$	−1.54324	−0.167388
$86$	0	0
$87$	−8.50319	−0.911638
$88$	0	0
$89$	16.9059	1.79202	0.896012	−	0.444031i	$-0.146452\pi$
$89$	16.9059	1.79202	0.896012	+	0.444031i	$0.146452\pi$
$90$	0	0
$91$	−0.518957	−0.0544015
$92$	0	0
$93$	7.00691	0.726583
$94$	0	0
$95$	−5.61957	−0.576555
$96$	0	0
$97$	−10.8771	−1.10440	−0.552202	−	0.833711i	$-0.686212\pi$
$97$	−10.8771	−1.10440	−0.552202	+	0.833711i	$0.686212\pi$
$98$	0	0
$99$	−7.64143	−0.767993
$100$	0	0
$101$	14.5634	1.44912	0.724558	−	0.689214i	$-0.242044\pi$
$101$	14.5634	1.44912	0.724558	+	0.689214i	$0.242044\pi$
$102$	0	0
$103$	3.02276	0.297842	0.148921	−	0.988849i	$-0.452420\pi$
$103$	3.02276	0.297842	0.148921	+	0.988849i	$0.452420\pi$
$104$	0	0
$105$	−2.61327	−0.255029
$106$	0	0
$107$	−0.813792	−0.0786722	−0.0393361	−	0.999226i	$-0.512524\pi$
$107$	−0.813792	−0.0786722	−0.0393361	+	0.999226i	$0.512524\pi$
$108$	0	0
$109$	−0.462071	−0.0442584	−0.0221292	−	0.999755i	$-0.507045\pi$
$109$	−0.462071	−0.0442584	−0.0221292	+	0.999755i	$0.507045\pi$
$110$	0	0
$111$	1.49007	0.141431
$112$	0	0
$113$	−1.13940	−0.107186	−0.0535929	−	0.998563i	$-0.517067\pi$
$113$	−1.13940	−0.107186	−0.0535929	+	0.998563i	$0.517067\pi$
$114$	0	0
$115$	−7.57256	−0.706145
$116$	0	0
$117$	1.00415	0.0928338
$118$	0	0
$119$	−0.538532	−0.0493671
$120$	0	0
$121$	8.15002	0.740911
$122$	0	0
$123$	−2.11703	−0.190887
$124$	0	0
$125$	−8.56543	−0.766115
$126$	0	0
$127$	−16.9069	−1.50024	−0.750122	−	0.661300i	$-0.770005\pi$
$127$	−16.9069	−1.50024	−0.750122	+	0.661300i	$0.770005\pi$
$128$	0	0
$129$	−13.4692	−1.18589
$130$	0	0
$131$	20.8815	1.82443	0.912213	−	0.409717i	$-0.134372\pi$
$131$	20.8815	1.82443	0.912213	+	0.409717i	$0.134372\pi$
$132$	0	0
$133$	−1.96101	−0.170041
$134$	0	0
$135$	13.7438	1.18288
$136$	0	0
$137$	7.16786	0.612392	0.306196	−	0.951969i	$-0.400944\pi$
$137$	7.16786	0.612392	0.306196	+	0.951969i	$0.400944\pi$
$138$	0	0
$139$	18.0675	1.53247	0.766233	−	0.642563i	$-0.222129\pi$
$139$	18.0675	1.53247	0.766233	+	0.642563i	$0.222129\pi$
$140$	0	0
$141$	−3.51217	−0.295778
$142$	0	0
$143$	−2.51648	−0.210439
$144$	0	0
$145$	19.6386	1.63089
$146$	0	0
$147$	6.92624	0.571267
$148$	0	0
$149$	−16.0522	−1.31505	−0.657525	−	0.753433i	$-0.728397\pi$
$149$	−16.0522	−1.31505	−0.657525	+	0.753433i	$0.728397\pi$
$150$	0	0
$151$	−7.16701	−0.583243	−0.291621	−	0.956534i	$-0.594195\pi$
$151$	−7.16701	−0.583243	−0.291621	+	0.956534i	$0.594195\pi$
$152$	0	0
$153$	1.04203	0.0842429
$154$	0	0
$155$	−16.1828	−1.29984
$156$	0	0
$157$	7.09601	0.566323	0.283162	−	0.959072i	$-0.408617\pi$
$157$	7.09601	0.566323	0.283162	+	0.959072i	$0.408617\pi$
$158$	0	0
$159$	−8.44474	−0.669712
$160$	0	0
$161$	−2.64253	−0.208261
$162$	0	0
$163$	6.86506	0.537713	0.268856	−	0.963180i	$-0.413354\pi$
$163$	6.86506	0.537713	0.268856	+	0.963180i	$0.413354\pi$
$164$	0	0
$165$	−12.6720	−0.986516
$166$	0	0
$167$	3.94201	0.305042	0.152521	−	0.988300i	$-0.451261\pi$
$167$	3.94201	0.305042	0.152521	+	0.988300i	$0.451261\pi$
$168$	0	0
$169$	−12.6693	−0.974563
$170$	0	0
$171$	3.79445	0.290169
$172$	0	0
$173$	−1.07684	−0.0818708	−0.0409354	−	0.999162i	$-0.513034\pi$
$173$	−1.07684	−0.0818708	−0.0409354	+	0.999162i	$0.513034\pi$
$174$	0	0
$175$	1.52324	0.115146
$176$	0	0
$177$	−3.47229	−0.260993
$178$	0	0
$179$	−3.15386	−0.235730	−0.117865	−	0.993030i	$-0.537605\pi$
$179$	−3.15386	−0.235730	−0.117865	+	0.993030i	$0.537605\pi$
$180$	0	0
$181$	−2.39515	−0.178030	−0.0890150	−	0.996030i	$-0.528372\pi$
$181$	−2.39515	−0.178030	−0.0890150	+	0.996030i	$0.528372\pi$
$182$	0	0
$183$	7.22442	0.534045
$184$	0	0
$185$	−3.44139	−0.253016
$186$	0	0
$187$	−2.61140	−0.190965
$188$	0	0
$189$	4.79605	0.348861
$190$	0	0
$191$	19.7728	1.43071	0.715354	−	0.698763i	$-0.246266\pi$
$191$	19.7728	1.43071	0.715354	+	0.698763i	$0.246266\pi$
$192$	0	0
$193$	−20.7986	−1.49712	−0.748558	−	0.663069i	$-0.769253\pi$
$193$	−20.7986	−1.49712	−0.748558	+	0.663069i	$0.769253\pi$
$194$	0	0
$195$	1.66521	0.119248
$196$	0	0
$197$	−4.37064	−0.311395	−0.155698	−	0.987805i	$-0.549763\pi$
$197$	−4.37064	−0.311395	−0.155698	+	0.987805i	$0.549763\pi$
$198$	0	0
$199$	15.2579	1.08160	0.540800	−	0.841151i	$-0.318121\pi$
$199$	15.2579	1.08160	0.540800	+	0.841151i	$0.318121\pi$
$200$	0	0
$201$	−15.9884	−1.12773
$202$	0	0
$203$	6.85311	0.480994
$204$	0	0
$205$	4.88940	0.341491
$206$	0	0
$207$	5.11315	0.355388
$208$	0	0
$209$	−9.50918	−0.657763
$210$	0	0
$211$	−5.10106	−0.351171	−0.175586	−	0.984464i	$-0.556182\pi$
$211$	−5.10106	−0.351171	−0.175586	+	0.984464i	$0.556182\pi$
$212$	0	0
$213$	0.0431837	0.00295890
$214$	0	0
$215$	31.1077	2.12153
$216$	0	0
$217$	−5.64719	−0.383356
$218$	0	0
$219$	9.43857	0.637800
$220$	0	0
$221$	0.343161	0.0230835
$222$	0	0
$223$	1.14578	0.0767271	0.0383636	−	0.999264i	$-0.487785\pi$
$223$	1.14578	0.0767271	0.0383636	+	0.999264i	$0.487785\pi$
$224$	0	0
$225$	−2.94737	−0.196492
$226$	0	0
$227$	17.4222	1.15635	0.578176	−	0.815912i	$-0.303765\pi$
$227$	17.4222	1.15635	0.578176	+	0.815912i	$0.303765\pi$
$228$	0	0
$229$	−15.4845	−1.02324	−0.511622	−	0.859211i	$-0.670955\pi$
$229$	−15.4845	−1.02324	−0.511622	+	0.859211i	$0.670955\pi$
$230$	0	0
$231$	−4.42205	−0.290950
$232$	0	0
$233$	13.9577	0.914398	0.457199	−	0.889364i	$-0.348853\pi$
$233$	13.9577	0.914398	0.457199	+	0.889364i	$0.348853\pi$
$234$	0	0
$235$	8.11155	0.529139
$236$	0	0
$237$	−7.10482	−0.461508
$238$	0	0
$239$	−14.0600	−0.909464	−0.454732	−	0.890628i	$-0.650265\pi$
$239$	−14.0600	−0.909464	−0.454732	+	0.890628i	$0.650265\pi$
$240$	0	0
$241$	−1.48455	−0.0956281	−0.0478141	−	0.998856i	$-0.515225\pi$
$241$	−1.48455	−0.0956281	−0.0478141	+	0.998856i	$0.515225\pi$
$242$	0	0
$243$	−15.1459	−0.971609
$244$	0	0
$245$	−15.9965	−1.02198
$246$	0	0
$247$	1.24959	0.0795094
$248$	0	0
$249$	0.438257	0.0277734
$250$	0	0
$251$	24.2182	1.52864	0.764320	−	0.644838i	$-0.223075\pi$
$251$	24.2182	1.52864	0.764320	+	0.644838i	$0.223075\pi$
$252$	0	0
$253$	−12.8139	−0.805606
$254$	0	0
$255$	1.72803	0.108213
$256$	0	0
$257$	−3.41223	−0.212849	−0.106425	−	0.994321i	$-0.533940\pi$
$257$	−3.41223	−0.212849	−0.106425	+	0.994321i	$0.533940\pi$
$258$	0	0
$259$	−1.20091	−0.0746211
$260$	0	0
$261$	−13.2604	−0.820796
$262$	0	0
$263$	−19.5252	−1.20398	−0.601988	−	0.798505i	$-0.705624\pi$
$263$	−19.5252	−1.20398	−0.601988	+	0.798505i	$0.705624\pi$
$264$	0	0
$265$	19.5036	1.19810
$266$	0	0
$267$	−18.9302	−1.15851
$268$	0	0
$269$	3.83710	0.233952	0.116976	−	0.993135i	$-0.462680\pi$
$269$	3.83710	0.233952	0.116976	+	0.993135i	$0.462680\pi$
$270$	0	0
$271$	19.0010	1.15423	0.577115	−	0.816663i	$-0.304178\pi$
$271$	19.0010	1.15423	0.577115	+	0.816663i	$0.304178\pi$
$272$	0	0
$273$	0.581096	0.0351695
$274$	0	0
$275$	7.38635	0.445413
$276$	0	0
$277$	27.0529	1.62545	0.812724	−	0.582649i	$-0.197984\pi$
$277$	27.0529	1.62545	0.812724	+	0.582649i	$0.197984\pi$
$278$	0	0
$279$	10.9270	0.654181
$280$	0	0
$281$	−16.2968	−0.972187	−0.486094	−	0.873907i	$-0.661579\pi$
$281$	−16.2968	−0.972187	−0.486094	+	0.873907i	$0.661579\pi$
$282$	0	0
$283$	−3.96846	−0.235900	−0.117950	−	0.993020i	$-0.537632\pi$
$283$	−3.96846	−0.235900	−0.117950	+	0.993020i	$0.537632\pi$
$284$	0	0
$285$	6.29245	0.372732
$286$	0	0
$287$	1.70621	0.100715
$288$	0	0
$289$	−16.6439	−0.979053
$290$	0	0
$291$	12.1795	0.713976
$292$	0	0
$293$	13.6453	0.797169	0.398584	−	0.917132i	$-0.369502\pi$
$293$	13.6453	0.797169	0.398584	+	0.917132i	$0.369502\pi$
$294$	0	0
$295$	8.01944	0.466910
$296$	0	0
$297$	23.2566	1.34948
$298$	0	0
$299$	1.68386	0.0973804
$300$	0	0
$301$	10.8554	0.625695
$302$	0	0
$303$	−16.3072	−0.936827
$304$	0	0
$305$	−16.6852	−0.955391
$306$	0	0
$307$	6.67414	0.380913	0.190457	−	0.981696i	$-0.439003\pi$
$307$	6.67414	0.380913	0.190457	+	0.981696i	$0.439003\pi$
$308$	0	0
$309$	−3.38470	−0.192549
$310$	0	0
$311$	25.8999	1.46865	0.734324	−	0.678800i	$-0.237500\pi$
$311$	25.8999	1.46865	0.734324	+	0.678800i	$0.237500\pi$
$312$	0	0
$313$	0.315952	0.0178587	0.00892934	−	0.999960i	$-0.497158\pi$
$313$	0.315952	0.0178587	0.00892934	+	0.999960i	$0.497158\pi$
$314$	0	0
$315$	−4.07528	−0.229616
$316$	0	0
$317$	20.4710	1.14977	0.574883	−	0.818235i	$-0.305047\pi$
$317$	20.4710	1.14977	0.574883	+	0.818235i	$0.305047\pi$
$318$	0	0
$319$	33.2315	1.86061
$320$	0	0
$321$	0.911235	0.0508602
$322$	0	0
$323$	1.29672	0.0721516
$324$	0	0
$325$	−0.970630	−0.0538409
$326$	0	0
$327$	0.517399	0.0286122
$328$	0	0
$329$	2.83062	0.156057
$330$	0	0
$331$	−21.3958	−1.17602	−0.588009	−	0.808854i	$-0.700088\pi$
$331$	−21.3958	−1.17602	−0.588009	+	0.808854i	$0.700088\pi$
$332$	0	0
$333$	2.32370	0.127338
$334$	0	0
$335$	36.9260	2.01748
$336$	0	0
$337$	−1.27334	−0.0693631	−0.0346816	−	0.999398i	$-0.511042\pi$
$337$	−1.27334	−0.0693631	−0.0346816	+	0.999398i	$0.511042\pi$
$338$	0	0
$339$	1.27583	0.0692936
$340$	0	0
$341$	−27.3838	−1.48292
$342$	0	0
$343$	−11.8993	−0.642503
$344$	0	0
$345$	8.47929	0.456510
$346$	0	0
$347$	29.4920	1.58321	0.791606	−	0.611032i	$-0.209245\pi$
$347$	29.4920	1.58321	0.791606	+	0.611032i	$0.209245\pi$
$348$	0	0
$349$	20.2847	1.08582	0.542908	−	0.839792i	$-0.317323\pi$
$349$	20.2847	1.08582	0.542908	+	0.839792i	$0.317323\pi$
$350$	0	0
$351$	−3.05612	−0.163124
$352$	0	0
$353$	4.73741	0.252147	0.126073	−	0.992021i	$-0.459762\pi$
$353$	4.73741	0.252147	0.126073	+	0.992021i	$0.459762\pi$
$354$	0	0
$355$	−0.0997350	−0.00529339
$356$	0	0
$357$	0.603015	0.0319149
$358$	0	0
$359$	18.8431	0.994500	0.497250	−	0.867607i	$-0.334343\pi$
$359$	18.8431	0.994500	0.497250	+	0.867607i	$0.334343\pi$
$360$	0	0
$361$	−14.2781	−0.751479
$362$	0	0
$363$	−9.12589	−0.478985
$364$	0	0
$365$	−21.7989	−1.14101
$366$	0	0
$367$	7.33106	0.382678	0.191339	−	0.981524i	$-0.438717\pi$
$367$	7.33106	0.382678	0.191339	+	0.981524i	$0.438717\pi$
$368$	0	0
$369$	−3.30142	−0.171865
$370$	0	0
$371$	6.80600	0.353350
$372$	0	0
$373$	1.48656	0.0769711	0.0384856	−	0.999259i	$-0.487747\pi$
$373$	1.48656	0.0769711	0.0384856	+	0.999259i	$0.487747\pi$
$374$	0	0
$375$	9.59104	0.495279
$376$	0	0
$377$	−4.36691	−0.224907
$378$	0	0
$379$	−14.3909	−0.739211	−0.369605	−	0.929189i	$-0.620507\pi$
$379$	−14.3909	−0.739211	−0.369605	+	0.929189i	$0.620507\pi$
$380$	0	0
$381$	18.9313	0.969880
$382$	0	0
$383$	20.2886	1.03670	0.518351	−	0.855168i	$-0.326546\pi$
$383$	20.2886	1.03670	0.518351	+	0.855168i	$0.326546\pi$
$384$	0	0
$385$	10.2130	0.520501
$386$	0	0
$387$	−21.0046	−1.06772
$388$	0	0
$389$	−3.07847	−0.156085	−0.0780423	−	0.996950i	$-0.524867\pi$
$389$	−3.07847	−0.156085	−0.0780423	+	0.996950i	$0.524867\pi$
$390$	0	0
$391$	1.74738	0.0883688
$392$	0	0
$393$	−23.3818	−1.17946
$394$	0	0
$395$	16.4090	0.825624
$396$	0	0
$397$	3.14807	0.157997	0.0789987	−	0.996875i	$-0.474828\pi$
$397$	3.14807	0.157997	0.0789987	+	0.996875i	$0.474828\pi$
$398$	0	0
$399$	2.19582	0.109929
$400$	0	0
$401$	13.4963	0.673975	0.336987	−	0.941509i	$-0.390592\pi$
$401$	13.4963	0.673975	0.336987	+	0.941509i	$0.390592\pi$
$402$	0	0
$403$	3.59848	0.179253
$404$	0	0
$405$	−1.84203	−0.0915312
$406$	0	0
$407$	−5.82336	−0.288653
$408$	0	0
$409$	37.7025	1.86427	0.932134	−	0.362113i	$-0.117944\pi$
$409$	37.7025	1.86427	0.932134	+	0.362113i	$0.117944\pi$
$410$	0	0
$411$	−8.02613	−0.395900
$412$	0	0
$413$	2.79848	0.137704
$414$	0	0
$415$	−1.01218	−0.0496858
$416$	0	0
$417$	−20.2309	−0.990711
$418$	0	0
$419$	−17.7428	−0.866791	−0.433395	−	0.901204i	$-0.642685\pi$
$419$	−17.7428	−0.866791	−0.433395	+	0.901204i	$0.642685\pi$
$420$	0	0
$421$	2.73192	0.133145	0.0665727	−	0.997782i	$-0.478794\pi$
$421$	2.73192	0.133145	0.0665727	+	0.997782i	$0.478794\pi$
$422$	0	0
$423$	−5.47709	−0.266305
$424$	0	0
$425$	−1.00724	−0.0488584
$426$	0	0
$427$	−5.82249	−0.281770
$428$	0	0
$429$	2.81780	0.136045
$430$	0	0
$431$	24.2444	1.16781	0.583906	−	0.811821i	$-0.301524\pi$
$431$	24.2444	1.16781	0.583906	+	0.811821i	$0.301524\pi$
$432$	0	0
$433$	−31.5299	−1.51523	−0.757616	−	0.652701i	$-0.773636\pi$
$433$	−31.5299	−1.51523	−0.757616	+	0.652701i	$0.773636\pi$
$434$	0	0
$435$	−21.9901	−1.05434
$436$	0	0
$437$	6.36292	0.304380
$438$	0	0
$439$	−6.94082	−0.331267	−0.165634	−	0.986187i	$-0.552967\pi$
$439$	−6.94082	−0.331267	−0.165634	+	0.986187i	$0.552967\pi$
$440$	0	0
$441$	10.8012	0.514342
$442$	0	0
$443$	13.2462	0.629346	0.314673	−	0.949200i	$-0.398105\pi$
$443$	13.2462	0.629346	0.314673	+	0.949200i	$0.398105\pi$
$444$	0	0
$445$	43.7203	2.07254
$446$	0	0
$447$	17.9743	0.850156
$448$	0	0
$449$	−1.25941	−0.0594354	−0.0297177	−	0.999558i	$-0.509461\pi$
$449$	−1.25941	−0.0594354	−0.0297177	+	0.999558i	$0.509461\pi$
$450$	0	0
$451$	8.27362	0.389590
$452$	0	0
$453$	8.02518	0.377056
$454$	0	0
$455$	−1.34207	−0.0629173
$456$	0	0
$457$	13.4998	0.631496	0.315748	−	0.948843i	$-0.397745\pi$
$457$	13.4998	0.631496	0.315748	+	0.948843i	$0.397745\pi$
$458$	0	0
$459$	−3.17139	−0.148028
$460$	0	0
$461$	−40.0618	−1.86586	−0.932931	−	0.360055i	$-0.882758\pi$
$461$	−40.0618	−1.86586	−0.932931	+	0.360055i	$0.882758\pi$
$462$	0	0
$463$	−11.5289	−0.535795	−0.267898	−	0.963447i	$-0.586329\pi$
$463$	−11.5289	−0.535795	−0.267898	+	0.963447i	$0.586329\pi$
$464$	0	0
$465$	18.1205	0.840320
$466$	0	0
$467$	−3.91315	−0.181079	−0.0905395	−	0.995893i	$-0.528859\pi$
$467$	−3.91315	−0.181079	−0.0905395	+	0.995893i	$0.528859\pi$
$468$	0	0
$469$	12.8858	0.595010
$470$	0	0
$471$	−7.94568	−0.366118
$472$	0	0
$473$	52.6391	2.42035
$474$	0	0
$475$	−3.66778	−0.168289
$476$	0	0
$477$	−13.1692	−0.602977
$478$	0	0
$479$	−16.8398	−0.769430	−0.384715	−	0.923035i	$-0.625700\pi$
$479$	−16.8398	−0.769430	−0.384715	+	0.923035i	$0.625700\pi$
$480$	0	0
$481$	0.765240	0.0348919
$482$	0	0
$483$	2.95895	0.134637
$484$	0	0
$485$	−28.1292	−1.27728
$486$	0	0
$487$	−6.23605	−0.282582	−0.141291	−	0.989968i	$-0.545125\pi$
$487$	−6.23605	−0.282582	−0.141291	+	0.989968i	$0.545125\pi$
$488$	0	0
$489$	−7.68707	−0.347621
$490$	0	0
$491$	−22.2462	−1.00396	−0.501979	−	0.864880i	$-0.667395\pi$
$491$	−22.2462	−1.00396	−0.501979	+	0.864880i	$0.667395\pi$
$492$	0	0
$493$	−4.53163	−0.204094
$494$	0	0
$495$	−19.7615	−0.888213
$496$	0	0
$497$	−0.0348037	−0.00156116
$498$	0	0
$499$	11.6587	0.521915	0.260958	−	0.965350i	$-0.415962\pi$
$499$	11.6587	0.521915	0.260958	+	0.965350i	$0.415962\pi$
$500$	0	0
$501$	−4.41403	−0.197204
$502$	0	0
$503$	−10.7361	−0.478699	−0.239350	−	0.970933i	$-0.576934\pi$
$503$	−10.7361	−0.478699	−0.239350	+	0.970933i	$0.576934\pi$
$504$	0	0
$505$	37.6624	1.67596
$506$	0	0
$507$	14.1863	0.630037
$508$	0	0
$509$	26.5127	1.17515	0.587577	−	0.809168i	$-0.300082\pi$
$509$	26.5127	1.17515	0.587577	+	0.809168i	$0.300082\pi$
$510$	0	0
$511$	−7.60698	−0.336513
$512$	0	0
$513$	−11.5483	−0.509872
$514$	0	0
$515$	7.81716	0.344465
$516$	0	0
$517$	13.7260	0.603669
$518$	0	0
$519$	1.20578	0.0529280
$520$	0	0
$521$	−42.1766	−1.84779	−0.923895	−	0.382645i	$-0.875013\pi$
$521$	−42.1766	−1.84779	−0.923895	+	0.382645i	$0.875013\pi$
$522$	0	0
$523$	14.5529	0.636355	0.318178	−	0.948031i	$-0.396929\pi$
$523$	14.5529	0.636355	0.318178	+	0.948031i	$0.396929\pi$
$524$	0	0
$525$	−1.70563	−0.0744397
$526$	0	0
$527$	3.73421	0.162665
$528$	0	0
$529$	−14.4257	−0.627206
$530$	0	0
$531$	−5.41489	−0.234986
$532$	0	0
$533$	−1.08723	−0.0470930
$534$	0	0
$535$	−2.10454	−0.0909874
$536$	0	0
$537$	3.53150	0.152395
$538$	0	0
$539$	−27.0686	−1.16593
$540$	0	0
$541$	−31.4976	−1.35419	−0.677094	−	0.735896i	$-0.736761\pi$
$541$	−31.4976	−1.35419	−0.677094	+	0.735896i	$0.736761\pi$
$542$	0	0
$543$	2.68194	0.115093
$544$	0	0
$545$	−1.19496	−0.0511864
$546$	0	0
$547$	41.2921	1.76552	0.882761	−	0.469822i	$-0.155682\pi$
$547$	41.2921	1.76552	0.882761	+	0.469822i	$0.155682\pi$
$548$	0	0
$549$	11.2662	0.480829
$550$	0	0
$551$	−16.5015	−0.702988
$552$	0	0
$553$	5.72610	0.243498
$554$	0	0
$555$	3.85346	0.163570
$556$	0	0
$557$	−7.51888	−0.318585	−0.159293	−	0.987231i	$-0.550921\pi$
$557$	−7.51888	−0.318585	−0.159293	+	0.987231i	$0.550921\pi$
$558$	0	0
$559$	−6.91724	−0.292568
$560$	0	0
$561$	2.92409	0.123455
$562$	0	0
$563$	−31.0697	−1.30943	−0.654715	−	0.755876i	$-0.727211\pi$
$563$	−31.0697	−1.30943	−0.654715	+	0.755876i	$0.727211\pi$
$564$	0	0
$565$	−2.94660	−0.123964
$566$	0	0
$567$	−0.642798	−0.0269950
$568$	0	0
$569$	23.7447	0.995428	0.497714	−	0.867341i	$-0.334173\pi$
$569$	23.7447	0.995428	0.497714	+	0.867341i	$0.334173\pi$
$570$	0	0
$571$	−24.6739	−1.03257	−0.516285	−	0.856417i	$-0.672685\pi$
$571$	−24.6739	−1.03257	−0.516285	+	0.856417i	$0.672685\pi$
$572$	0	0
$573$	−22.1403	−0.924926
$574$	0	0
$575$	−4.94246	−0.206115
$576$	0	0
$577$	28.2250	1.17502	0.587511	−	0.809216i	$-0.300108\pi$
$577$	28.2250	1.17502	0.587511	+	0.809216i	$0.300108\pi$
$578$	0	0
$579$	23.2890	0.967858
$580$	0	0
$581$	−0.353211	−0.0146537
$582$	0	0
$583$	33.0031	1.36685
$584$	0	0
$585$	2.59683	0.107366
$586$	0	0
$587$	30.4799	1.25804	0.629020	−	0.777389i	$-0.283456\pi$
$587$	30.4799	1.25804	0.629020	+	0.777389i	$0.283456\pi$
$588$	0	0
$589$	13.5978	0.560287
$590$	0	0
$591$	4.89398	0.201311
$592$	0	0
$593$	6.05503	0.248650	0.124325	−	0.992242i	$-0.460323\pi$
$593$	6.05503	0.248650	0.124325	+	0.992242i	$0.460323\pi$
$594$	0	0
$595$	−1.39270	−0.0570949
$596$	0	0
$597$	−17.0848	−0.699235
$598$	0	0
$599$	32.6773	1.33516	0.667580	−	0.744538i	$-0.267330\pi$
$599$	32.6773	1.33516	0.667580	+	0.744538i	$0.267330\pi$
$600$	0	0
$601$	26.4431	1.07864	0.539318	−	0.842103i	$-0.318682\pi$
$601$	26.4431	1.07864	0.539318	+	0.842103i	$0.318682\pi$
$602$	0	0
$603$	−24.9332	−1.01536
$604$	0	0
$605$	21.0767	0.856891
$606$	0	0
$607$	38.2300	1.55171	0.775853	−	0.630914i	$-0.217320\pi$
$607$	38.2300	1.55171	0.775853	+	0.630914i	$0.217320\pi$
$608$	0	0
$609$	−7.67369	−0.310954
$610$	0	0
$611$	−1.80372	−0.0729705
$612$	0	0
$613$	24.7333	0.998967	0.499484	−	0.866323i	$-0.333523\pi$
$613$	24.7333	0.998967	0.499484	+	0.866323i	$0.333523\pi$
$614$	0	0
$615$	−5.47485	−0.220767
$616$	0	0
$617$	4.40815	0.177466	0.0887328	−	0.996055i	$-0.471718\pi$
$617$	4.40815	0.177466	0.0887328	+	0.996055i	$0.471718\pi$
$618$	0	0
$619$	39.2190	1.57634	0.788172	−	0.615455i	$-0.211028\pi$
$619$	39.2190	1.57634	0.788172	+	0.615455i	$0.211028\pi$
$620$	0	0
$621$	−15.5618	−0.624473
$622$	0	0
$623$	15.2567	0.611247
$624$	0	0
$625$	−30.5905	−1.22362
$626$	0	0
$627$	10.6478	0.425232
$628$	0	0
$629$	0.794105	0.0316630
$630$	0	0
$631$	−39.6179	−1.57716	−0.788582	−	0.614929i	$-0.789185\pi$
$631$	−39.6179	−1.57716	−0.788582	+	0.614929i	$0.789185\pi$
$632$	0	0
$633$	5.71185	0.227026
$634$	0	0
$635$	−43.7228	−1.73509
$636$	0	0
$637$	3.55705	0.140935
$638$	0	0
$639$	0.0673431	0.00266405
$640$	0	0
$641$	14.8069	0.584836	0.292418	−	0.956291i	$-0.405540\pi$
$641$	14.8069	0.584836	0.292418	+	0.956291i	$0.405540\pi$
$642$	0	0
$643$	−11.9402	−0.470877	−0.235439	−	0.971889i	$-0.575653\pi$
$643$	−11.9402	−0.470877	−0.235439	+	0.971889i	$0.575653\pi$
$644$	0	0
$645$	−34.8325	−1.37153
$646$	0	0
$647$	−31.6854	−1.24568	−0.622841	−	0.782349i	$-0.714022\pi$
$647$	−31.6854	−1.24568	−0.622841	+	0.782349i	$0.714022\pi$
$648$	0	0
$649$	13.5701	0.532674
$650$	0	0
$651$	6.32338	0.247833
$652$	0	0
$653$	32.5995	1.27572	0.637859	−	0.770153i	$-0.279820\pi$
$653$	32.5995	1.27572	0.637859	+	0.770153i	$0.279820\pi$
$654$	0	0
$655$	54.0016	2.11002
$656$	0	0
$657$	14.7191	0.574245
$658$	0	0
$659$	−14.5998	−0.568728	−0.284364	−	0.958716i	$-0.591782\pi$
$659$	−14.5998	−0.568728	−0.284364	+	0.958716i	$0.591782\pi$
$660$	0	0
$661$	14.5568	0.566194	0.283097	−	0.959091i	$-0.408638\pi$
$661$	14.5568	0.566194	0.283097	+	0.959091i	$0.408638\pi$
$662$	0	0
$663$	−0.384250	−0.0149231
$664$	0	0
$665$	−5.07137	−0.196659
$666$	0	0
$667$	−22.2363	−0.860995
$668$	0	0
$669$	−1.28297	−0.0496027
$670$	0	0
$671$	−28.2339	−1.08996
$672$	0	0
$673$	−44.6087	−1.71954	−0.859770	−	0.510681i	$-0.829393\pi$
$673$	−44.6087	−1.71954	−0.859770	+	0.510681i	$0.829393\pi$
$674$	0	0
$675$	8.97029	0.345267
$676$	0	0
$677$	20.9920	0.806790	0.403395	−	0.915026i	$-0.367830\pi$
$677$	20.9920	0.806790	0.403395	+	0.915026i	$0.367830\pi$
$678$	0	0
$679$	−9.81603	−0.376705
$680$	0	0
$681$	−19.5083	−0.747560
$682$	0	0
$683$	19.9420	0.763060	0.381530	−	0.924357i	$-0.375397\pi$
$683$	19.9420	0.763060	0.381530	+	0.924357i	$0.375397\pi$
$684$	0	0
$685$	18.5368	0.708254
$686$	0	0
$687$	17.3386	0.661508
$688$	0	0
$689$	−4.33689	−0.165222
$690$	0	0
$691$	24.3922	0.927923	0.463962	−	0.885855i	$-0.346428\pi$
$691$	24.3922	0.927923	0.463962	+	0.885855i	$0.346428\pi$
$692$	0	0
$693$	−6.89600	−0.261957
$694$	0	0
$695$	46.7243	1.77235
$696$	0	0
$697$	−1.12824	−0.0427350
$698$	0	0
$699$	−15.6290	−0.591142
$700$	0	0
$701$	4.31333	0.162912	0.0814562	−	0.996677i	$-0.474043\pi$
$701$	4.31333	0.162912	0.0814562	+	0.996677i	$0.474043\pi$
$702$	0	0
$703$	2.89166	0.109061
$704$	0	0
$705$	−9.08282	−0.342079
$706$	0	0
$707$	13.1428	0.494284
$708$	0	0
$709$	−17.4259	−0.654442	−0.327221	−	0.944948i	$-0.606112\pi$
$709$	−17.4259	−0.654442	−0.327221	+	0.944948i	$0.606112\pi$
$710$	0	0
$711$	−11.0797	−0.415520
$712$	0	0
$713$	18.3235	0.686220
$714$	0	0
$715$	−6.50786	−0.243380
$716$	0	0
$717$	15.7435	0.587951
$718$	0	0
$719$	−16.3402	−0.609386	−0.304693	−	0.952451i	$-0.598554\pi$
$719$	−16.3402	−0.609386	−0.304693	+	0.952451i	$0.598554\pi$
$720$	0	0
$721$	2.72789	0.101592
$722$	0	0
$723$	1.66231	0.0618218
$724$	0	0
$725$	12.8177	0.476038
$726$	0	0
$727$	−15.8673	−0.588487	−0.294244	−	0.955730i	$-0.595068\pi$
$727$	−15.8673	−0.588487	−0.294244	+	0.955730i	$0.595068\pi$
$728$	0	0
$729$	19.0963	0.707270
$730$	0	0
$731$	−7.17815	−0.265494
$732$	0	0
$733$	19.7548	0.729662	0.364831	−	0.931074i	$-0.381127\pi$
$733$	19.7548	0.729662	0.364831	+	0.931074i	$0.381127\pi$
$734$	0	0
$735$	17.9119	0.660691
$736$	0	0
$737$	62.4845	2.30165
$738$	0	0
$739$	−38.2977	−1.40880	−0.704402	−	0.709802i	$-0.748785\pi$
$739$	−38.2977	−1.40880	−0.704402	+	0.709802i	$0.748785\pi$
$740$	0	0
$741$	−1.39921	−0.0514014
$742$	0	0
$743$	46.4053	1.70244	0.851222	−	0.524805i	$-0.175862\pi$
$743$	46.4053	1.70244	0.851222	+	0.524805i	$0.175862\pi$
$744$	0	0
$745$	−41.5126	−1.52090
$746$	0	0
$747$	0.683443	0.0250059
$748$	0	0
$749$	−0.734405	−0.0268346
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−27.1181	−0.988237
$754$	0	0
$755$	−18.5346	−0.674542
$756$	0	0
$757$	25.7400	0.935537	0.467768	−	0.883851i	$-0.345058\pi$
$757$	25.7400	0.935537	0.467768	+	0.883851i	$0.345058\pi$
$758$	0	0
$759$	14.3483	0.520809
$760$	0	0
$761$	−9.66520	−0.350363	−0.175182	−	0.984536i	$-0.556051\pi$
$761$	−9.66520	−0.350363	−0.175182	+	0.984536i	$0.556051\pi$
$762$	0	0
$763$	−0.416995	−0.0150962
$764$	0	0
$765$	2.69478	0.0974301
$766$	0	0
$767$	−1.78323	−0.0643888
$768$	0	0
$769$	−24.9551	−0.899904	−0.449952	−	0.893053i	$-0.648559\pi$
$769$	−24.9551	−0.899904	−0.449952	+	0.893053i	$0.648559\pi$
$770$	0	0
$771$	3.82081	0.137603
$772$	0	0
$773$	22.9736	0.826302	0.413151	−	0.910663i	$-0.364428\pi$
$773$	22.9736	0.826302	0.413151	+	0.910663i	$0.364428\pi$
$774$	0	0
$775$	−10.5622	−0.379406
$776$	0	0
$777$	1.34471	0.0482411
$778$	0	0
$779$	−4.10837	−0.147198
$780$	0	0
$781$	−0.168767	−0.00603896
$782$	0	0
$783$	40.3577	1.44227
$784$	0	0
$785$	18.3510	0.654974
$786$	0	0
$787$	20.4129	0.727642	0.363821	−	0.931469i	$-0.381472\pi$
$787$	20.4129	0.727642	0.363821	+	0.931469i	$0.381472\pi$
$788$	0	0
$789$	21.8631	0.778348
$790$	0	0
$791$	−1.02825	−0.0365604
$792$	0	0
$793$	3.71018	0.131752
$794$	0	0
$795$	−21.8389	−0.774547
$796$	0	0
$797$	43.3559	1.53575	0.767873	−	0.640603i	$-0.221315\pi$
$797$	43.3559	1.53575	0.767873	+	0.640603i	$0.221315\pi$
$798$	0	0
$799$	−1.87175	−0.0662178
$800$	0	0
$801$	−29.5209	−1.04307
$802$	0	0
$803$	−36.8871	−1.30172
$804$	0	0
$805$	−6.83385	−0.240862
$806$	0	0
$807$	−4.29655	−0.151246
$808$	0	0
$809$	13.1857	0.463586	0.231793	−	0.972765i	$-0.425541\pi$
$809$	13.1857	0.463586	0.231793	+	0.972765i	$0.425541\pi$
$810$	0	0
$811$	21.4263	0.752379	0.376190	−	0.926543i	$-0.377234\pi$
$811$	21.4263	0.752379	0.376190	+	0.926543i	$0.377234\pi$
$812$	0	0
$813$	−21.2762	−0.746189
$814$	0	0
$815$	17.7537	0.621885
$816$	0	0
$817$	−26.1386	−0.914473
$818$	0	0
$819$	0.906195	0.0316650
$820$	0	0
$821$	−32.8733	−1.14729	−0.573643	−	0.819105i	$-0.694470\pi$
$821$	−32.8733	−1.14729	−0.573643	+	0.819105i	$0.694470\pi$
$822$	0	0
$823$	−50.9541	−1.77615	−0.888073	−	0.459702i	$-0.847956\pi$
$823$	−50.9541	−1.77615	−0.888073	+	0.459702i	$0.847956\pi$
$824$	0	0
$825$	−8.27078	−0.287952
$826$	0	0
$827$	−38.3336	−1.33299	−0.666494	−	0.745510i	$-0.732206\pi$
$827$	−38.3336	−1.33299	−0.666494	+	0.745510i	$0.732206\pi$
$828$	0	0
$829$	−3.68082	−0.127840	−0.0639200	−	0.997955i	$-0.520360\pi$
$829$	−3.68082	−0.127840	−0.0639200	+	0.997955i	$0.520360\pi$
$830$	0	0
$831$	−30.2921	−1.05082
$832$	0	0
$833$	3.69122	0.127893
$834$	0	0
$835$	10.1944	0.352793
$836$	0	0
$837$	−33.2561	−1.14950
$838$	0	0
$839$	−20.0364	−0.691733	−0.345867	−	0.938284i	$-0.612415\pi$
$839$	−20.0364	−0.691733	−0.345867	+	0.938284i	$0.612415\pi$
$840$	0	0
$841$	28.6674	0.988531
$842$	0	0
$843$	18.2482	0.628501
$844$	0	0
$845$	−32.7641	−1.12712
$846$	0	0
$847$	7.35497	0.252720
$848$	0	0
$849$	4.44364	0.152505
$850$	0	0
$851$	3.89661	0.133574
$852$	0	0
$853$	−37.7757	−1.29342	−0.646708	−	0.762737i	$-0.723855\pi$
$853$	−37.7757	−1.29342	−0.646708	+	0.762737i	$0.723855\pi$
$854$	0	0
$855$	9.81281	0.335591
$856$	0	0
$857$	−47.5978	−1.62591	−0.812954	−	0.582327i	$-0.802142\pi$
$857$	−47.5978	−1.62591	−0.812954	+	0.582327i	$0.802142\pi$
$858$	0	0
$859$	−40.0285	−1.36576	−0.682878	−	0.730533i	$-0.739272\pi$
$859$	−40.0285	−1.36576	−0.682878	+	0.730533i	$0.739272\pi$
$860$	0	0
$861$	−1.91051	−0.0651102
$862$	0	0
$863$	−13.6102	−0.463296	−0.231648	−	0.972800i	$-0.574412\pi$
$863$	−13.6102	−0.463296	−0.231648	+	0.972800i	$0.574412\pi$
$864$	0	0
$865$	−2.78482	−0.0946867
$866$	0	0
$867$	18.6368	0.632939
$868$	0	0
$869$	27.7665	0.941913
$870$	0	0
$871$	−8.21101	−0.278219
$872$	0	0
$873$	18.9934	0.642831
$874$	0	0
$875$	−7.72986	−0.261317
$876$	0	0
$877$	33.1313	1.11877	0.559383	−	0.828909i	$-0.311038\pi$
$877$	33.1313	1.11877	0.559383	+	0.828909i	$0.311038\pi$
$878$	0	0
$879$	−15.2792	−0.515355
$880$	0	0
$881$	23.0817	0.777642	0.388821	−	0.921313i	$-0.372883\pi$
$881$	23.0817	0.777642	0.388821	+	0.921313i	$0.372883\pi$
$882$	0	0
$883$	−42.8623	−1.44243	−0.721215	−	0.692711i	$-0.756416\pi$
$883$	−42.8623	−1.44243	−0.721215	+	0.692711i	$0.756416\pi$
$884$	0	0
$885$	−8.97968	−0.301849
$886$	0	0
$887$	30.8296	1.03516	0.517578	−	0.855636i	$-0.326834\pi$
$887$	30.8296	1.03516	0.517578	+	0.855636i	$0.326834\pi$
$888$	0	0
$889$	−15.2576	−0.511723
$890$	0	0
$891$	−3.11700	−0.104423
$892$	0	0
$893$	−6.81581	−0.228082
$894$	0	0
$895$	−8.15618	−0.272631
$896$	0	0
$897$	−1.88549	−0.0629546
$898$	0	0
$899$	−47.5199	−1.58488
$900$	0	0
$901$	−4.50048	−0.149933
$902$	0	0
$903$	−12.1552	−0.404501
$904$	0	0
$905$	−6.19409	−0.205898
$906$	0	0
$907$	−13.0926	−0.434731	−0.217366	−	0.976090i	$-0.569746\pi$
$907$	−13.0926	−0.434731	−0.217366	+	0.976090i	$0.569746\pi$
$908$	0	0
$909$	−25.4305	−0.843475
$910$	0	0
$911$	−51.5173	−1.70684	−0.853421	−	0.521222i	$-0.825477\pi$
$911$	−51.5173	−1.70684	−0.853421	+	0.521222i	$0.825477\pi$
$912$	0	0
$913$	−1.71276	−0.0566841
$914$	0	0
$915$	18.6830	0.617642
$916$	0	0
$917$	18.8445	0.622300
$918$	0	0
$919$	7.44528	0.245597	0.122799	−	0.992432i	$-0.460813\pi$
$919$	7.44528	0.245597	0.122799	+	0.992432i	$0.460813\pi$
$920$	0	0
$921$	−7.47329	−0.246253
$922$	0	0
$923$	0.0221775	0.000729980 0
$924$	0	0
$925$	−2.24613	−0.0738521
$926$	0	0
$927$	−5.27830	−0.173362
$928$	0	0
$929$	16.2817	0.534184	0.267092	−	0.963671i	$-0.413937\pi$
$929$	16.2817	0.534184	0.267092	+	0.963671i	$0.413937\pi$
$930$	0	0
$931$	13.4412	0.440519
$932$	0	0
$933$	−29.0011	−0.949453
$934$	0	0
$935$	−6.75333	−0.220858
$936$	0	0
$937$	0.880037	0.0287496	0.0143748	−	0.999897i	$-0.495424\pi$
$937$	0.880037	0.0287496	0.0143748	+	0.999897i	$0.495424\pi$
$938$	0	0
$939$	−0.353784	−0.0115453
$940$	0	0
$941$	−58.4200	−1.90444	−0.952218	−	0.305418i	$-0.901204\pi$
$941$	−58.4200	−1.90444	−0.952218	+	0.305418i	$0.901204\pi$
$942$	0	0
$943$	−5.53617	−0.180282
$944$	0	0
$945$	12.4031	0.403471
$946$	0	0
$947$	26.2451	0.852851	0.426425	−	0.904523i	$-0.359773\pi$
$947$	26.2451	0.852851	0.426425	+	0.904523i	$0.359773\pi$
$948$	0	0
$949$	4.84728	0.157349
$950$	0	0
$951$	−22.9222	−0.743303
$952$	0	0
$953$	−8.16547	−0.264505	−0.132253	−	0.991216i	$-0.542221\pi$
$953$	−8.16547	−0.264505	−0.132253	+	0.991216i	$0.542221\pi$
$954$	0	0
$955$	51.1343	1.65467
$956$	0	0
$957$	−37.2106	−1.20285
$958$	0	0
$959$	6.46863	0.208883
$960$	0	0
$961$	8.15793	0.263159
$962$	0	0
$963$	1.42103	0.0457921
$964$	0	0
$965$	−53.7872	−1.73147
$966$	0	0
$967$	7.06111	0.227070	0.113535	−	0.993534i	$-0.463783\pi$
$967$	7.06111	0.227070	0.113535	+	0.993534i	$0.463783\pi$
$968$	0	0
$969$	−1.45199	−0.0466447
$970$	0	0
$971$	−37.6549	−1.20840	−0.604202	−	0.796831i	$-0.706508\pi$
$971$	−37.6549	−1.20840	−0.604202	+	0.796831i	$0.706508\pi$
$972$	0	0
$973$	16.3050	0.522714
$974$	0	0
$975$	1.08685	0.0348071
$976$	0	0
$977$	20.1421	0.644403	0.322202	−	0.946671i	$-0.395577\pi$
$977$	20.1421	0.644403	0.322202	+	0.946671i	$0.395577\pi$
$978$	0	0
$979$	73.9815	2.36446
$980$	0	0
$981$	0.806861	0.0257611
$982$	0	0
$983$	−47.9888	−1.53061	−0.765303	−	0.643670i	$-0.777411\pi$
$983$	−47.9888	−1.53061	−0.765303	+	0.643670i	$0.777411\pi$
$984$	0	0
$985$	−11.3029	−0.360140
$986$	0	0
$987$	−3.16956	−0.100888
$988$	0	0
$989$	−35.2226	−1.12002
$990$	0	0
$991$	−47.8194	−1.51903	−0.759516	−	0.650488i	$-0.774564\pi$
$991$	−47.8194	−1.51903	−0.759516	+	0.650488i	$0.774564\pi$
$992$	0	0
$993$	23.9577	0.760274
$994$	0	0
$995$	39.4583	1.25091
$996$	0	0
$997$	−16.5907	−0.525432	−0.262716	−	0.964873i	$-0.584618\pi$
$997$	−16.5907	−0.525432	−0.262716	+	0.964873i	$0.584618\pi$
$998$	0	0
$999$	−7.07213	−0.223752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.15	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.15	✓	49	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.11974 q^{3} +2.58610 q^{5} +0.902449 q^{7} -1.74619 q^{9} +O(q^{10})\)	\(q-1.11974 q^{3} +2.58610 q^{5} +0.902449 q^{7} -1.74619 q^{9} +4.37607 q^{11} -0.575054 q^{13} -2.89575 q^{15} -0.596745 q^{17} -2.17299 q^{19} -1.01051 q^{21} -2.92818 q^{23} +1.68789 q^{25} +5.31449 q^{27} +7.59391 q^{29} -6.25763 q^{31} -4.90006 q^{33} +2.33382 q^{35} -1.33073 q^{37} +0.643910 q^{39} +1.89065 q^{41} +12.0288 q^{43} -4.51580 q^{45} +3.13660 q^{47} -6.18559 q^{49} +0.668199 q^{51} +7.54171 q^{53} +11.3169 q^{55} +2.43318 q^{57} +3.10098 q^{59} -6.45188 q^{61} -1.57584 q^{63} -1.48715 q^{65} +14.2787 q^{67} +3.27880 q^{69} -0.0385659 q^{71} -8.42926 q^{73} -1.89000 q^{75} +3.94918 q^{77} +6.34507 q^{79} -0.712282 q^{81} -0.391392 q^{83} -1.54324 q^{85} -8.50319 q^{87} +16.9059 q^{89} -0.518957 q^{91} +7.00691 q^{93} -5.61957 q^{95} -10.8771 q^{97} -7.64143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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