LMFDB - Embedded newform 6008.2.a.c.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.87868 q^{3} -2.95953 q^{5} -5.20423 q^{7} +5.28682 q^{9} +O(q^{10})$	$q-2.87868 q^{3} -2.95953 q^{5} -5.20423 q^{7} +5.28682 q^{9} +6.14319 q^{11} -2.39873 q^{13} +8.51956 q^{15} -6.97884 q^{17} -3.26012 q^{19} +14.9813 q^{21} -2.38374 q^{23} +3.75883 q^{25} -6.58303 q^{27} -7.00797 q^{29} -0.758665 q^{31} -17.6843 q^{33} +15.4021 q^{35} -2.09923 q^{37} +6.90517 q^{39} +11.4353 q^{41} +4.49758 q^{43} -15.6465 q^{45} -3.55075 q^{47} +20.0840 q^{49} +20.0899 q^{51} +5.35427 q^{53} -18.1810 q^{55} +9.38484 q^{57} +2.22991 q^{59} +0.141552 q^{61} -27.5138 q^{63} +7.09910 q^{65} +9.78018 q^{67} +6.86204 q^{69} +11.1016 q^{71} +6.27245 q^{73} -10.8205 q^{75} -31.9706 q^{77} -9.72696 q^{79} +3.09001 q^{81} +14.9108 q^{83} +20.6541 q^{85} +20.1737 q^{87} -6.94358 q^{89} +12.4835 q^{91} +2.18396 q^{93} +9.64842 q^{95} +1.53402 q^{97} +32.4780 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * 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
$3$	−2.87868	−1.66201	−0.831004	−	0.556266i	$-0.812234\pi$
$3$	−2.87868	−1.66201	−0.831004	+	0.556266i	$0.812234\pi$
$4$	0	0
$5$	−2.95953	−1.32354	−0.661772	−	0.749706i	$-0.730195\pi$
$5$	−2.95953	−1.32354	−0.661772	+	0.749706i	$0.730195\pi$
$6$	0	0
$7$	−5.20423	−1.96701	−0.983507	−	0.180868i	$-0.942109\pi$
$7$	−5.20423	−1.96701	−0.983507	+	0.180868i	$0.942109\pi$
$8$	0	0
$9$	5.28682	1.76227
$10$	0	0
$11$	6.14319	1.85224	0.926121	−	0.377226i	$-0.123122\pi$
$11$	6.14319	1.85224	0.926121	+	0.377226i	$0.123122\pi$
$12$	0	0
$13$	−2.39873	−0.665287	−0.332643	−	0.943053i	$-0.607941\pi$
$13$	−2.39873	−0.665287	−0.332643	+	0.943053i	$0.607941\pi$
$14$	0	0
$15$	8.51956	2.19974
$16$	0	0
$17$	−6.97884	−1.69262	−0.846308	−	0.532694i	$-0.821180\pi$
$17$	−6.97884	−1.69262	−0.846308	+	0.532694i	$0.821180\pi$
$18$	0	0
$19$	−3.26012	−0.747922	−0.373961	−	0.927444i	$-0.622001\pi$
$19$	−3.26012	−0.747922	−0.373961	+	0.927444i	$0.622001\pi$
$20$	0	0
$21$	14.9813	3.26920
$22$	0	0
$23$	−2.38374	−0.497044	−0.248522	−	0.968626i	$-0.579945\pi$
$23$	−2.38374	−0.497044	−0.248522	+	0.968626i	$0.579945\pi$
$24$	0	0
$25$	3.75883	0.751766
$26$	0	0
$27$	−6.58303	−1.26690
$28$	0	0
$29$	−7.00797	−1.30135	−0.650673	−	0.759358i	$-0.725513\pi$
$29$	−7.00797	−1.30135	−0.650673	+	0.759358i	$0.725513\pi$
$30$	0	0
$31$	−0.758665	−0.136260	−0.0681301	−	0.997676i	$-0.521703\pi$
$31$	−0.758665	−0.136260	−0.0681301	+	0.997676i	$0.521703\pi$
$32$	0	0
$33$	−17.6843	−3.07844
$34$	0	0
$35$	15.4021	2.60343
$36$	0	0
$37$	−2.09923	−0.345111	−0.172555	−	0.985000i	$-0.555202\pi$
$37$	−2.09923	−0.345111	−0.172555	+	0.985000i	$0.555202\pi$
$38$	0	0
$39$	6.90517	1.10571
$40$	0	0
$41$	11.4353	1.78590	0.892950	−	0.450156i	$-0.148632\pi$
$41$	11.4353	1.78590	0.892950	+	0.450156i	$0.148632\pi$
$42$	0	0
$43$	4.49758	0.685874	0.342937	−	0.939358i	$-0.388578\pi$
$43$	4.49758	0.685874	0.342937	+	0.939358i	$0.388578\pi$
$44$	0	0
$45$	−15.6465	−2.33244
$46$	0	0
$47$	−3.55075	−0.517930	−0.258965	−	0.965887i	$-0.583381\pi$
$47$	−3.55075	−0.517930	−0.258965	+	0.965887i	$0.583381\pi$
$48$	0	0
$49$	20.0840	2.86915
$50$	0	0
$51$	20.0899	2.81314
$52$	0	0
$53$	5.35427	0.735466	0.367733	−	0.929931i	$-0.380134\pi$
$53$	5.35427	0.735466	0.367733	+	0.929931i	$0.380134\pi$
$54$	0	0
$55$	−18.1810	−2.45152
$56$	0	0
$57$	9.38484	1.24305
$58$	0	0
$59$	2.22991	0.290310	0.145155	−	0.989409i	$-0.453632\pi$
$59$	2.22991	0.290310	0.145155	+	0.989409i	$0.453632\pi$
$60$	0	0
$61$	0.141552	0.0181238	0.00906191	−	0.999959i	$-0.497115\pi$
$61$	0.141552	0.0181238	0.00906191	+	0.999959i	$0.497115\pi$
$62$	0	0
$63$	−27.5138	−3.46642
$64$	0	0
$65$	7.09910	0.880536
$66$	0	0
$67$	9.78018	1.19484	0.597420	−	0.801929i	$-0.296193\pi$
$67$	9.78018	1.19484	0.597420	+	0.801929i	$0.296193\pi$
$68$	0	0
$69$	6.86204	0.826092
$70$	0	0
$71$	11.1016	1.31752	0.658758	−	0.752355i	$-0.271082\pi$
$71$	11.1016	1.31752	0.658758	+	0.752355i	$0.271082\pi$
$72$	0	0
$73$	6.27245	0.734134	0.367067	−	0.930194i	$-0.380362\pi$
$73$	6.27245	0.734134	0.367067	+	0.930194i	$0.380362\pi$
$74$	0	0
$75$	−10.8205	−1.24944
$76$	0	0
$77$	−31.9706	−3.64339
$78$	0	0
$79$	−9.72696	−1.09437	−0.547184	−	0.837012i	$-0.684300\pi$
$79$	−9.72696	−1.09437	−0.547184	+	0.837012i	$0.684300\pi$
$80$	0	0
$81$	3.09001	0.343334
$82$	0	0
$83$	14.9108	1.63667	0.818336	−	0.574740i	$-0.194897\pi$
$83$	14.9108	1.63667	0.818336	+	0.574740i	$0.194897\pi$
$84$	0	0
$85$	20.6541	2.24025
$86$	0	0
$87$	20.1737	2.16285
$88$	0	0
$89$	−6.94358	−0.736018	−0.368009	−	0.929822i	$-0.619960\pi$
$89$	−6.94358	−0.736018	−0.368009	+	0.929822i	$0.619960\pi$
$90$	0	0
$91$	12.4835	1.30863
$92$	0	0
$93$	2.18396	0.226466
$94$	0	0
$95$	9.64842	0.989907
$96$	0	0
$97$	1.53402	0.155756	0.0778782	−	0.996963i	$-0.475185\pi$
$97$	1.53402	0.155756	0.0778782	+	0.996963i	$0.475185\pi$
$98$	0	0
$99$	32.4780	3.26416
$100$	0	0
$101$	−18.1389	−1.80489	−0.902446	−	0.430804i	$-0.858230\pi$
$101$	−18.1389	−1.80489	−0.902446	+	0.430804i	$0.858230\pi$
$102$	0	0
$103$	16.9857	1.67365	0.836826	−	0.547469i	$-0.184408\pi$
$103$	16.9857	1.67365	0.836826	+	0.547469i	$0.184408\pi$
$104$	0	0
$105$	−44.3378	−4.32692
$106$	0	0
$107$	1.77717	0.171805	0.0859027	−	0.996304i	$-0.472623\pi$
$107$	1.77717	0.171805	0.0859027	+	0.996304i	$0.472623\pi$
$108$	0	0
$109$	−1.50140	−0.143808	−0.0719040	−	0.997412i	$-0.522908\pi$
$109$	−1.50140	−0.143808	−0.0719040	+	0.997412i	$0.522908\pi$
$110$	0	0
$111$	6.04301	0.573577
$112$	0	0
$113$	4.06669	0.382562	0.191281	−	0.981535i	$-0.438736\pi$
$113$	4.06669	0.382562	0.191281	+	0.981535i	$0.438736\pi$
$114$	0	0
$115$	7.05476	0.657860
$116$	0	0
$117$	−12.6816	−1.17242
$118$	0	0
$119$	36.3195	3.32940
$120$	0	0
$121$	26.7388	2.43080
$122$	0	0
$123$	−32.9187	−2.96818
$124$	0	0
$125$	3.67328	0.328548
$126$	0	0
$127$	10.4137	0.924064	0.462032	−	0.886863i	$-0.347120\pi$
$127$	10.4137	0.924064	0.462032	+	0.886863i	$0.347120\pi$
$128$	0	0
$129$	−12.9471	−1.13993
$130$	0	0
$131$	−0.502702	−0.0439213	−0.0219606	−	0.999759i	$-0.506991\pi$
$131$	−0.502702	−0.0439213	−0.0219606	+	0.999759i	$0.506991\pi$
$132$	0	0
$133$	16.9664	1.47117
$134$	0	0
$135$	19.4827	1.67680
$136$	0	0
$137$	−19.3648	−1.65444	−0.827222	−	0.561875i	$-0.810080\pi$
$137$	−19.3648	−1.65444	−0.827222	+	0.561875i	$0.810080\pi$
$138$	0	0
$139$	8.60448	0.729823	0.364911	−	0.931042i	$-0.381099\pi$
$139$	8.60448	0.729823	0.364911	+	0.931042i	$0.381099\pi$
$140$	0	0
$141$	10.2215	0.860804
$142$	0	0
$143$	−14.7358	−1.23227
$144$	0	0
$145$	20.7403	1.72239
$146$	0	0
$147$	−57.8156	−4.76855
$148$	0	0
$149$	12.4251	1.01790	0.508951	−	0.860795i	$-0.330033\pi$
$149$	12.4251	1.01790	0.508951	+	0.860795i	$0.330033\pi$
$150$	0	0
$151$	23.9090	1.94569	0.972843	−	0.231465i	$-0.0743520\pi$
$151$	23.9090	1.94569	0.972843	+	0.231465i	$0.0743520\pi$
$152$	0	0
$153$	−36.8958	−2.98285
$154$	0	0
$155$	2.24529	0.180346
$156$	0	0
$157$	−8.33298	−0.665044	−0.332522	−	0.943095i	$-0.607900\pi$
$157$	−8.33298	−0.665044	−0.332522	+	0.943095i	$0.607900\pi$
$158$	0	0
$159$	−15.4133	−1.22235
$160$	0	0
$161$	12.4055	0.977694
$162$	0	0
$163$	2.33044	0.182534	0.0912672	−	0.995826i	$-0.470908\pi$
$163$	2.33044	0.182534	0.0912672	+	0.995826i	$0.470908\pi$
$164$	0	0
$165$	52.3373	4.07445
$166$	0	0
$167$	−1.25022	−0.0967453	−0.0483726	−	0.998829i	$-0.515403\pi$
$167$	−1.25022	−0.0967453	−0.0483726	+	0.998829i	$0.515403\pi$
$168$	0	0
$169$	−7.24612	−0.557394
$170$	0	0
$171$	−17.2356	−1.31804
$172$	0	0
$173$	−21.0098	−1.59735	−0.798674	−	0.601764i	$-0.794465\pi$
$173$	−21.0098	−1.59735	−0.798674	+	0.601764i	$0.794465\pi$
$174$	0	0
$175$	−19.5618	−1.47874
$176$	0	0
$177$	−6.41921	−0.482498
$178$	0	0
$179$	−12.2178	−0.913205	−0.456602	−	0.889671i	$-0.650934\pi$
$179$	−12.2178	−0.913205	−0.456602	+	0.889671i	$0.650934\pi$
$180$	0	0
$181$	−16.1641	−1.20146	−0.600732	−	0.799450i	$-0.705124\pi$
$181$	−16.1641	−1.20146	−0.600732	+	0.799450i	$0.705124\pi$
$182$	0	0
$183$	−0.407482	−0.0301219
$184$	0	0
$185$	6.21273	0.456769
$186$	0	0
$187$	−42.8723	−3.13514
$188$	0	0
$189$	34.2596	2.49202
$190$	0	0
$191$	−3.69370	−0.267267	−0.133633	−	0.991031i	$-0.542664\pi$
$191$	−3.69370	−0.267267	−0.133633	+	0.991031i	$0.542664\pi$
$192$	0	0
$193$	−23.1037	−1.66304	−0.831519	−	0.555497i	$-0.812528\pi$
$193$	−23.1037	−1.66304	−0.831519	+	0.555497i	$0.812528\pi$
$194$	0	0
$195$	−20.4361	−1.46346
$196$	0	0
$197$	−22.9397	−1.63438	−0.817192	−	0.576365i	$-0.804470\pi$
$197$	−22.9397	−1.63438	−0.817192	+	0.576365i	$0.804470\pi$
$198$	0	0
$199$	−11.7828	−0.835261	−0.417631	−	0.908617i	$-0.637139\pi$
$199$	−11.7828	−0.835261	−0.417631	+	0.908617i	$0.637139\pi$
$200$	0	0
$201$	−28.1540	−1.98583
$202$	0	0
$203$	36.4711	2.55977
$204$	0	0
$205$	−33.8432	−2.36371
$206$	0	0
$207$	−12.6024	−0.875928
$208$	0	0
$209$	−20.0275	−1.38533
$210$	0	0
$211$	−20.3295	−1.39954	−0.699769	−	0.714369i	$-0.746714\pi$
$211$	−20.3295	−1.39954	−0.699769	+	0.714369i	$0.746714\pi$
$212$	0	0
$213$	−31.9579	−2.18972
$214$	0	0
$215$	−13.3107	−0.907784
$216$	0	0
$217$	3.94827	0.268026
$218$	0	0
$219$	−18.0564	−1.22014
$220$	0	0
$221$	16.7403	1.12608
$222$	0	0
$223$	3.34965	0.224309	0.112155	−	0.993691i	$-0.464225\pi$
$223$	3.34965	0.224309	0.112155	+	0.993691i	$0.464225\pi$
$224$	0	0
$225$	19.8723	1.32482
$226$	0	0
$227$	−17.2583	−1.14547	−0.572737	−	0.819739i	$-0.694118\pi$
$227$	−17.2583	−1.14547	−0.572737	+	0.819739i	$0.694118\pi$
$228$	0	0
$229$	−29.6575	−1.95982	−0.979912	−	0.199432i	$-0.936090\pi$
$229$	−29.6575	−1.95982	−0.979912	+	0.199432i	$0.936090\pi$
$230$	0	0
$231$	92.0332	6.05534
$232$	0	0
$233$	15.4129	1.00973	0.504866	−	0.863198i	$-0.331542\pi$
$233$	15.4129	1.00973	0.504866	+	0.863198i	$0.331542\pi$
$234$	0	0
$235$	10.5086	0.685502
$236$	0	0
$237$	28.0008	1.81885
$238$	0	0
$239$	17.7547	1.14845	0.574227	−	0.818696i	$-0.305303\pi$
$239$	17.7547	1.14845	0.574227	+	0.818696i	$0.305303\pi$
$240$	0	0
$241$	11.6927	0.753196	0.376598	−	0.926377i	$-0.377094\pi$
$241$	11.6927	0.753196	0.376598	+	0.926377i	$0.377094\pi$
$242$	0	0
$243$	10.8539	0.696281
$244$	0	0
$245$	−59.4393	−3.79744
$246$	0	0
$247$	7.82012	0.497582
$248$	0	0
$249$	−42.9234	−2.72016
$250$	0	0
$251$	3.71300	0.234363	0.117181	−	0.993111i	$-0.462614\pi$
$251$	3.71300	0.234363	0.117181	+	0.993111i	$0.462614\pi$
$252$	0	0
$253$	−14.6438	−0.920647
$254$	0	0
$255$	−59.4566	−3.72332
$256$	0	0
$257$	−8.92193	−0.556535	−0.278267	−	0.960504i	$-0.589760\pi$
$257$	−8.92193	−0.556535	−0.278267	+	0.960504i	$0.589760\pi$
$258$	0	0
$259$	10.9249	0.678838
$260$	0	0
$261$	−37.0499	−2.29333
$262$	0	0
$263$	18.0746	1.11453	0.557263	−	0.830336i	$-0.311852\pi$
$263$	18.0746	1.11453	0.557263	+	0.830336i	$0.311852\pi$
$264$	0	0
$265$	−15.8461	−0.973421
$266$	0	0
$267$	19.9884	1.22327
$268$	0	0
$269$	−4.46239	−0.272077	−0.136038	−	0.990704i	$-0.543437\pi$
$269$	−4.46239	−0.272077	−0.136038	+	0.990704i	$0.543437\pi$
$270$	0	0
$271$	18.9490	1.15107	0.575534	−	0.817778i	$-0.304794\pi$
$271$	18.9490	1.15107	0.575534	+	0.817778i	$0.304794\pi$
$272$	0	0
$273$	−35.9361	−2.17495
$274$	0	0
$275$	23.0912	1.39245
$276$	0	0
$277$	22.8343	1.37198	0.685991	−	0.727610i	$-0.259369\pi$
$277$	22.8343	1.37198	0.685991	+	0.727610i	$0.259369\pi$
$278$	0	0
$279$	−4.01093	−0.240128
$280$	0	0
$281$	24.7821	1.47837	0.739187	−	0.673501i	$-0.235210\pi$
$281$	24.7821	1.47837	0.739187	+	0.673501i	$0.235210\pi$
$282$	0	0
$283$	−6.41297	−0.381212	−0.190606	−	0.981667i	$-0.561045\pi$
$283$	−6.41297	−0.381212	−0.190606	+	0.981667i	$0.561045\pi$
$284$	0	0
$285$	−27.7747	−1.64523
$286$	0	0
$287$	−59.5121	−3.51289
$288$	0	0
$289$	31.7041	1.86495
$290$	0	0
$291$	−4.41597	−0.258868
$292$	0	0
$293$	24.0739	1.40641	0.703207	−	0.710985i	$-0.251751\pi$
$293$	24.0739	1.40641	0.703207	+	0.710985i	$0.251751\pi$
$294$	0	0
$295$	−6.59950	−0.384238
$296$	0	0
$297$	−40.4408	−2.34661
$298$	0	0
$299$	5.71794	0.330677
$300$	0	0
$301$	−23.4064	−1.34912
$302$	0	0
$303$	52.2163	2.99975
$304$	0	0
$305$	−0.418926	−0.0239877
$306$	0	0
$307$	12.8142	0.731343	0.365672	−	0.930744i	$-0.380839\pi$
$307$	12.8142	0.731343	0.365672	+	0.930744i	$0.380839\pi$
$308$	0	0
$309$	−48.8965	−2.78162
$310$	0	0
$311$	19.1150	1.08391	0.541957	−	0.840406i	$-0.317684\pi$
$311$	19.1150	1.08391	0.541957	+	0.840406i	$0.317684\pi$
$312$	0	0
$313$	9.90376	0.559793	0.279897	−	0.960030i	$-0.409700\pi$
$313$	9.90376	0.559793	0.279897	+	0.960030i	$0.409700\pi$
$314$	0	0
$315$	81.4281	4.58795
$316$	0	0
$317$	4.86510	0.273251	0.136626	−	0.990623i	$-0.456374\pi$
$317$	4.86510	0.273251	0.136626	+	0.990623i	$0.456374\pi$
$318$	0	0
$319$	−43.0513	−2.41041
$320$	0	0
$321$	−5.11590	−0.285542
$322$	0	0
$323$	22.7518	1.26594
$324$	0	0
$325$	−9.01640	−0.500140
$326$	0	0
$327$	4.32206	0.239010
$328$	0	0
$329$	18.4789	1.01878
$330$	0	0
$331$	−22.0500	−1.21198	−0.605988	−	0.795474i	$-0.707222\pi$
$331$	−22.0500	−1.21198	−0.605988	+	0.795474i	$0.707222\pi$
$332$	0	0
$333$	−11.0982	−0.608179
$334$	0	0
$335$	−28.9448	−1.58142
$336$	0	0
$337$	4.45030	0.242423	0.121212	−	0.992627i	$-0.461322\pi$
$337$	4.45030	0.242423	0.121212	+	0.992627i	$0.461322\pi$
$338$	0	0
$339$	−11.7067	−0.635822
$340$	0	0
$341$	−4.66063	−0.252387
$342$	0	0
$343$	−68.0924	−3.67664
$344$	0	0
$345$	−20.3084	−1.09337
$346$	0	0
$347$	−2.72389	−0.146226	−0.0731129	−	0.997324i	$-0.523293\pi$
$347$	−2.72389	−0.146226	−0.0731129	+	0.997324i	$0.523293\pi$
$348$	0	0
$349$	6.58142	0.352295	0.176148	−	0.984364i	$-0.443636\pi$
$349$	6.58142	0.352295	0.176148	+	0.984364i	$0.443636\pi$
$350$	0	0
$351$	15.7909	0.842855
$352$	0	0
$353$	−8.07764	−0.429929	−0.214965	−	0.976622i	$-0.568964\pi$
$353$	−8.07764	−0.429929	−0.214965	+	0.976622i	$0.568964\pi$
$354$	0	0
$355$	−32.8555	−1.74379
$356$	0	0
$357$	−104.552	−5.53349
$358$	0	0
$359$	−23.4296	−1.23657	−0.618283	−	0.785956i	$-0.712171\pi$
$359$	−23.4296	−1.23657	−0.618283	+	0.785956i	$0.712171\pi$
$360$	0	0
$361$	−8.37165	−0.440613
$362$	0	0
$363$	−76.9726	−4.04001
$364$	0	0
$365$	−18.5635	−0.971659
$366$	0	0
$367$	30.0289	1.56750	0.783748	−	0.621079i	$-0.213305\pi$
$367$	30.0289	1.56750	0.783748	+	0.621079i	$0.213305\pi$
$368$	0	0
$369$	60.4566	3.14724
$370$	0	0
$371$	−27.8649	−1.44667
$372$	0	0
$373$	3.18264	0.164791	0.0823954	−	0.996600i	$-0.473743\pi$
$373$	3.18264	0.164791	0.0823954	+	0.996600i	$0.473743\pi$
$374$	0	0
$375$	−10.5742	−0.546050
$376$	0	0
$377$	16.8102	0.865769
$378$	0	0
$379$	12.3665	0.635224	0.317612	−	0.948221i	$-0.397119\pi$
$379$	12.3665	0.635224	0.317612	+	0.948221i	$0.397119\pi$
$380$	0	0
$381$	−29.9777	−1.53580
$382$	0	0
$383$	4.69727	0.240019	0.120010	−	0.992773i	$-0.461707\pi$
$383$	4.69727	0.240019	0.120010	+	0.992773i	$0.461707\pi$
$384$	0	0
$385$	94.6180	4.82218
$386$	0	0
$387$	23.7779	1.20870
$388$	0	0
$389$	1.63472	0.0828837	0.0414419	−	0.999141i	$-0.486805\pi$
$389$	1.63472	0.0828837	0.0414419	+	0.999141i	$0.486805\pi$
$390$	0	0
$391$	16.6357	0.841306
$392$	0	0
$393$	1.44712	0.0729975
$394$	0	0
$395$	28.7873	1.44844
$396$	0	0
$397$	22.3910	1.12377	0.561885	−	0.827215i	$-0.310076\pi$
$397$	22.3910	1.12377	0.561885	+	0.827215i	$0.310076\pi$
$398$	0	0
$399$	−48.8409	−2.44510
$400$	0	0
$401$	23.7401	1.18552	0.592762	−	0.805378i	$-0.298037\pi$
$401$	23.7401	1.18552	0.592762	+	0.805378i	$0.298037\pi$
$402$	0	0
$403$	1.81983	0.0906521
$404$	0	0
$405$	−9.14497	−0.454417
$406$	0	0
$407$	−12.8959	−0.639228
$408$	0	0
$409$	−12.5805	−0.622067	−0.311033	−	0.950399i	$-0.600675\pi$
$409$	−12.5805	−0.622067	−0.311033	+	0.950399i	$0.600675\pi$
$410$	0	0
$411$	55.7451	2.74970
$412$	0	0
$413$	−11.6050	−0.571044
$414$	0	0
$415$	−44.1290	−2.16621
$416$	0	0
$417$	−24.7696	−1.21297
$418$	0	0
$419$	−28.0547	−1.37056	−0.685280	−	0.728280i	$-0.740320\pi$
$419$	−28.0547	−1.37056	−0.685280	+	0.728280i	$0.740320\pi$
$420$	0	0
$421$	4.27019	0.208116	0.104058	−	0.994571i	$-0.466817\pi$
$421$	4.27019	0.208116	0.104058	+	0.994571i	$0.466817\pi$
$422$	0	0
$423$	−18.7722	−0.912734
$424$	0	0
$425$	−26.2323	−1.27245
$426$	0	0
$427$	−0.736667	−0.0356498
$428$	0	0
$429$	42.4198	2.04805
$430$	0	0
$431$	6.75131	0.325199	0.162600	−	0.986692i	$-0.448012\pi$
$431$	6.75131	0.325199	0.162600	+	0.986692i	$0.448012\pi$
$432$	0	0
$433$	5.28031	0.253756	0.126878	−	0.991918i	$-0.459504\pi$
$433$	5.28031	0.253756	0.126878	+	0.991918i	$0.459504\pi$
$434$	0	0
$435$	−59.7048	−2.86262
$436$	0	0
$437$	7.77127	0.371750
$438$	0	0
$439$	21.0066	1.00259	0.501296	−	0.865276i	$-0.332857\pi$
$439$	21.0066	1.00259	0.501296	+	0.865276i	$0.332857\pi$
$440$	0	0
$441$	106.181	5.05622
$442$	0	0
$443$	−22.5202	−1.06997	−0.534984	−	0.844862i	$-0.679683\pi$
$443$	−22.5202	−1.06997	−0.534984	+	0.844862i	$0.679683\pi$
$444$	0	0
$445$	20.5498	0.974152
$446$	0	0
$447$	−35.7679	−1.69176
$448$	0	0
$449$	−4.30091	−0.202972	−0.101486	−	0.994837i	$-0.532360\pi$
$449$	−4.30091	−0.202972	−0.101486	+	0.994837i	$0.532360\pi$
$450$	0	0
$451$	70.2495	3.30792
$452$	0	0
$453$	−68.8264	−3.23375
$454$	0	0
$455$	−36.9454	−1.73203
$456$	0	0
$457$	−3.26970	−0.152950	−0.0764749	−	0.997072i	$-0.524367\pi$
$457$	−3.26970	−0.152950	−0.0764749	+	0.997072i	$0.524367\pi$
$458$	0	0
$459$	45.9419	2.14438
$460$	0	0
$461$	−6.39298	−0.297751	−0.148875	−	0.988856i	$-0.547565\pi$
$461$	−6.39298	−0.297751	−0.148875	+	0.988856i	$0.547565\pi$
$462$	0	0
$463$	−31.7265	−1.47445	−0.737227	−	0.675645i	$-0.763865\pi$
$463$	−31.7265	−1.47445	−0.737227	+	0.675645i	$0.763865\pi$
$464$	0	0
$465$	−6.46349	−0.299737
$466$	0	0
$467$	−15.1159	−0.699481	−0.349740	−	0.936847i	$-0.613730\pi$
$467$	−15.1159	−0.699481	−0.349740	+	0.936847i	$0.613730\pi$
$468$	0	0
$469$	−50.8983	−2.35027
$470$	0	0
$471$	23.9880	1.10531
$472$	0	0
$473$	27.6295	1.27040
$474$	0	0
$475$	−12.2542	−0.562262
$476$	0	0
$477$	28.3071	1.29609
$478$	0	0
$479$	19.4624	0.889261	0.444631	−	0.895714i	$-0.353335\pi$
$479$	19.4624	0.889261	0.444631	+	0.895714i	$0.353335\pi$
$480$	0	0
$481$	5.03547	0.229597
$482$	0	0
$483$	−35.7116	−1.62494
$484$	0	0
$485$	−4.53999	−0.206150
$486$	0	0
$487$	−2.10857	−0.0955483	−0.0477742	−	0.998858i	$-0.515213\pi$
$487$	−2.10857	−0.0955483	−0.0477742	+	0.998858i	$0.515213\pi$
$488$	0	0
$489$	−6.70861	−0.303374
$490$	0	0
$491$	22.4327	1.01237	0.506186	−	0.862424i	$-0.331055\pi$
$491$	22.4327	1.01237	0.506186	+	0.862424i	$0.331055\pi$
$492$	0	0
$493$	48.9074	2.20268
$494$	0	0
$495$	−96.1195	−4.32025
$496$	0	0
$497$	−57.7752	−2.59157
$498$	0	0
$499$	6.25312	0.279928	0.139964	−	0.990157i	$-0.455301\pi$
$499$	6.25312	0.279928	0.139964	+	0.990157i	$0.455301\pi$
$500$	0	0
$501$	3.59900	0.160791
$502$	0	0
$503$	−19.6189	−0.874764	−0.437382	−	0.899276i	$-0.644094\pi$
$503$	−19.6189	−0.874764	−0.437382	+	0.899276i	$0.644094\pi$
$504$	0	0
$505$	53.6828	2.38885
$506$	0	0
$507$	20.8593	0.926393
$508$	0	0
$509$	33.5794	1.48838	0.744189	−	0.667969i	$-0.232836\pi$
$509$	33.5794	1.48838	0.744189	+	0.667969i	$0.232836\pi$
$510$	0	0
$511$	−32.6433	−1.44405
$512$	0	0
$513$	21.4614	0.947546
$514$	0	0
$515$	−50.2698	−2.21515
$516$	0	0
$517$	−21.8129	−0.959331
$518$	0	0
$519$	60.4807	2.65481
$520$	0	0
$521$	−18.9860	−0.831793	−0.415897	−	0.909412i	$-0.636532\pi$
$521$	−18.9860	−0.831793	−0.415897	+	0.909412i	$0.636532\pi$
$522$	0	0
$523$	−43.1524	−1.88692	−0.943461	−	0.331484i	$-0.892451\pi$
$523$	−43.1524	−1.88692	−0.943461	+	0.331484i	$0.892451\pi$
$524$	0	0
$525$	56.3123	2.45767
$526$	0	0
$527$	5.29460	0.230636
$528$	0	0
$529$	−17.3178	−0.752947
$530$	0	0
$531$	11.7891	0.511606
$532$	0	0
$533$	−27.4302	−1.18814
$534$	0	0
$535$	−5.25959	−0.227392
$536$	0	0
$537$	35.1713	1.51775
$538$	0	0
$539$	123.380	5.31436
$540$	0	0
$541$	19.5773	0.841694	0.420847	−	0.907132i	$-0.361733\pi$
$541$	19.5773	0.841694	0.420847	+	0.907132i	$0.361733\pi$
$542$	0	0
$543$	46.5312	1.99684
$544$	0	0
$545$	4.44344	0.190336
$546$	0	0
$547$	−5.26947	−0.225306	−0.112653	−	0.993634i	$-0.535935\pi$
$547$	−5.26947	−0.225306	−0.112653	+	0.993634i	$0.535935\pi$
$548$	0	0
$549$	0.748358	0.0319391
$550$	0	0
$551$	22.8468	0.973305
$552$	0	0
$553$	50.6214	2.15264
$554$	0	0
$555$	−17.8845	−0.759154
$556$	0	0
$557$	−10.6269	−0.450278	−0.225139	−	0.974327i	$-0.572284\pi$
$557$	−10.6269	−0.450278	−0.225139	+	0.974327i	$0.572284\pi$
$558$	0	0
$559$	−10.7884	−0.456303
$560$	0	0
$561$	123.416	5.21062
$562$	0	0
$563$	−24.8323	−1.04656	−0.523279	−	0.852162i	$-0.675291\pi$
$563$	−24.8323	−1.04656	−0.523279	+	0.852162i	$0.675291\pi$
$564$	0	0
$565$	−12.0355	−0.506338
$566$	0	0
$567$	−16.0811	−0.675343
$568$	0	0
$569$	31.7567	1.33131	0.665654	−	0.746260i	$-0.268153\pi$
$569$	31.7567	1.33131	0.665654	+	0.746260i	$0.268153\pi$
$570$	0	0
$571$	−23.8410	−0.997716	−0.498858	−	0.866684i	$-0.666247\pi$
$571$	−23.8410	−0.997716	−0.498858	+	0.866684i	$0.666247\pi$
$572$	0	0
$573$	10.6330	0.444200
$574$	0	0
$575$	−8.96008	−0.373661
$576$	0	0
$577$	3.64984	0.151945	0.0759724	−	0.997110i	$-0.475794\pi$
$577$	3.64984	0.151945	0.0759724	+	0.997110i	$0.475794\pi$
$578$	0	0
$579$	66.5081	2.76398
$580$	0	0
$581$	−77.5992	−3.21936
$582$	0	0
$583$	32.8923	1.36226
$584$	0	0
$585$	37.5317	1.55174
$586$	0	0
$587$	−16.6398	−0.686796	−0.343398	−	0.939190i	$-0.611578\pi$
$587$	−16.6398	−0.686796	−0.343398	+	0.939190i	$0.611578\pi$
$588$	0	0
$589$	2.47334	0.101912
$590$	0	0
$591$	66.0361	2.71636
$592$	0	0
$593$	22.9050	0.940595	0.470298	−	0.882508i	$-0.344147\pi$
$593$	22.9050	0.940595	0.470298	+	0.882508i	$0.344147\pi$
$594$	0	0
$595$	−107.489	−4.40661
$596$	0	0
$597$	33.9190	1.38821
$598$	0	0
$599$	16.9793	0.693756	0.346878	−	0.937910i	$-0.387242\pi$
$599$	16.9793	0.693756	0.346878	+	0.937910i	$0.387242\pi$
$600$	0	0
$601$	−0.718245	−0.0292978	−0.0146489	−	0.999893i	$-0.504663\pi$
$601$	−0.718245	−0.0292978	−0.0146489	+	0.999893i	$0.504663\pi$
$602$	0	0
$603$	51.7060	2.10563
$604$	0	0
$605$	−79.1344	−3.21727
$606$	0	0
$607$	−37.6227	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$607$	−37.6227	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$608$	0	0
$609$	−104.989	−4.25436
$610$	0	0
$611$	8.51727	0.344572
$612$	0	0
$613$	3.00485	0.121365	0.0606824	−	0.998157i	$-0.480672\pi$
$613$	3.00485	0.121365	0.0606824	+	0.998157i	$0.480672\pi$
$614$	0	0
$615$	97.4240	3.92852
$616$	0	0
$617$	15.1426	0.609618	0.304809	−	0.952414i	$-0.401407\pi$
$617$	15.1426	0.609618	0.304809	+	0.952414i	$0.401407\pi$
$618$	0	0
$619$	9.26279	0.372303	0.186151	−	0.982521i	$-0.440399\pi$
$619$	9.26279	0.372303	0.186151	+	0.982521i	$0.440399\pi$
$620$	0	0
$621$	15.6922	0.629708
$622$	0	0
$623$	36.1360	1.44776
$624$	0	0
$625$	−29.6653	−1.18661
$626$	0	0
$627$	57.6529	2.30243
$628$	0	0
$629$	14.6502	0.584140
$630$	0	0
$631$	−14.0848	−0.560706	−0.280353	−	0.959897i	$-0.590452\pi$
$631$	−14.0848	−0.560706	−0.280353	+	0.959897i	$0.590452\pi$
$632$	0	0
$633$	58.5221	2.32604
$634$	0	0
$635$	−30.8196	−1.22304
$636$	0	0
$637$	−48.1761	−1.90881
$638$	0	0
$639$	58.6921	2.32182
$640$	0	0
$641$	24.9767	0.986521	0.493260	−	0.869882i	$-0.335805\pi$
$641$	24.9767	0.986521	0.493260	+	0.869882i	$0.335805\pi$
$642$	0	0
$643$	−44.2936	−1.74677	−0.873385	−	0.487031i	$-0.838080\pi$
$643$	−44.2936	−1.74677	−0.873385	+	0.487031i	$0.838080\pi$
$644$	0	0
$645$	38.3174	1.50874
$646$	0	0
$647$	23.8271	0.936739	0.468370	−	0.883533i	$-0.344842\pi$
$647$	23.8271	0.936739	0.468370	+	0.883533i	$0.344842\pi$
$648$	0	0
$649$	13.6988	0.537724
$650$	0	0
$651$	−11.3658	−0.445462
$652$	0	0
$653$	−1.86629	−0.0730335	−0.0365167	−	0.999333i	$-0.511626\pi$
$653$	−1.86629	−0.0730335	−0.0365167	+	0.999333i	$0.511626\pi$
$654$	0	0
$655$	1.48776	0.0581317
$656$	0	0
$657$	33.1613	1.29375
$658$	0	0
$659$	36.4081	1.41826	0.709129	−	0.705079i	$-0.249088\pi$
$659$	36.4081	1.41826	0.709129	+	0.705079i	$0.249088\pi$
$660$	0	0
$661$	−4.58593	−0.178372	−0.0891860	−	0.996015i	$-0.528427\pi$
$661$	−4.58593	−0.178372	−0.0891860	+	0.996015i	$0.528427\pi$
$662$	0	0
$663$	−48.1901	−1.87155
$664$	0	0
$665$	−50.2126	−1.94716
$666$	0	0
$667$	16.7052	0.646827
$668$	0	0
$669$	−9.64259	−0.372804
$670$	0	0
$671$	0.869578	0.0335697
$672$	0	0
$673$	−4.75817	−0.183414	−0.0917070	−	0.995786i	$-0.529232\pi$
$673$	−4.75817	−0.183414	−0.0917070	+	0.995786i	$0.529232\pi$
$674$	0	0
$675$	−24.7445	−0.952416
$676$	0	0
$677$	−4.13898	−0.159074	−0.0795369	−	0.996832i	$-0.525344\pi$
$677$	−4.13898	−0.159074	−0.0795369	+	0.996832i	$0.525344\pi$
$678$	0	0
$679$	−7.98341	−0.306375
$680$	0	0
$681$	49.6812	1.90379
$682$	0	0
$683$	−3.56141	−0.136274	−0.0681368	−	0.997676i	$-0.521705\pi$
$683$	−3.56141	−0.136274	−0.0681368	+	0.997676i	$0.521705\pi$
$684$	0	0
$685$	57.3107	2.18973
$686$	0	0
$687$	85.3746	3.25724
$688$	0	0
$689$	−12.8434	−0.489296
$690$	0	0
$691$	−4.31875	−0.164293	−0.0821465	−	0.996620i	$-0.526178\pi$
$691$	−4.31875	−0.164293	−0.0821465	+	0.996620i	$0.526178\pi$
$692$	0	0
$693$	−169.023	−6.42065
$694$	0	0
$695$	−25.4652	−0.965952
$696$	0	0
$697$	−79.8053	−3.02284
$698$	0	0
$699$	−44.3688	−1.67818
$700$	0	0
$701$	−19.9622	−0.753962	−0.376981	−	0.926221i	$-0.623038\pi$
$701$	−19.9622	−0.753962	−0.376981	+	0.926221i	$0.623038\pi$
$702$	0	0
$703$	6.84372	0.258116
$704$	0	0
$705$	−30.2508	−1.13931
$706$	0	0
$707$	94.3992	3.55025
$708$	0	0
$709$	−43.8589	−1.64715	−0.823577	−	0.567204i	$-0.808025\pi$
$709$	−43.8589	−1.64715	−0.823577	+	0.567204i	$0.808025\pi$
$710$	0	0
$711$	−51.4247	−1.92858
$712$	0	0
$713$	1.80846	0.0677274
$714$	0	0
$715$	43.6112	1.63097
$716$	0	0
$717$	−51.1100	−1.90874
$718$	0	0
$719$	−31.6081	−1.17878	−0.589391	−	0.807848i	$-0.700632\pi$
$719$	−31.6081	−1.17878	−0.589391	+	0.807848i	$0.700632\pi$
$720$	0	0
$721$	−88.3976	−3.29210
$722$	0	0
$723$	−33.6597	−1.25182
$724$	0	0
$725$	−26.3418	−0.978308
$726$	0	0
$727$	−19.9187	−0.738744	−0.369372	−	0.929282i	$-0.620427\pi$
$727$	−19.9187	−0.738744	−0.369372	+	0.929282i	$0.620427\pi$
$728$	0	0
$729$	−40.5151	−1.50056
$730$	0	0
$731$	−31.3878	−1.16092
$732$	0	0
$733$	23.8035	0.879203	0.439602	−	0.898193i	$-0.355120\pi$
$733$	23.8035	0.879203	0.439602	+	0.898193i	$0.355120\pi$
$734$	0	0
$735$	171.107	6.31138
$736$	0	0
$737$	60.0815	2.21313
$738$	0	0
$739$	28.9742	1.06583	0.532916	−	0.846168i	$-0.321096\pi$
$739$	28.9742	1.06583	0.532916	+	0.846168i	$0.321096\pi$
$740$	0	0
$741$	−22.5117	−0.826986
$742$	0	0
$743$	6.29508	0.230944	0.115472	−	0.993311i	$-0.463162\pi$
$743$	6.29508	0.230944	0.115472	+	0.993311i	$0.463162\pi$
$744$	0	0
$745$	−36.7724	−1.34724
$746$	0	0
$747$	78.8307	2.88426
$748$	0	0
$749$	−9.24879	−0.337944
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−10.6886	−0.389513
$754$	0	0
$755$	−70.7594	−2.57520
$756$	0	0
$757$	3.36636	0.122352	0.0611762	−	0.998127i	$-0.480515\pi$
$757$	3.36636	0.122352	0.0611762	+	0.998127i	$0.480515\pi$
$758$	0	0
$759$	42.1548	1.53012
$760$	0	0
$761$	−39.0864	−1.41688	−0.708440	−	0.705771i	$-0.750601\pi$
$761$	−39.0864	−1.41688	−0.708440	+	0.705771i	$0.750601\pi$
$762$	0	0
$763$	7.81363	0.282873
$764$	0	0
$765$	109.194	3.94793
$766$	0	0
$767$	−5.34895	−0.193139
$768$	0	0
$769$	28.9413	1.04365	0.521824	−	0.853053i	$-0.325252\pi$
$769$	28.9413	1.04365	0.521824	+	0.853053i	$0.325252\pi$
$770$	0	0
$771$	25.6834	0.924966
$772$	0	0
$773$	31.4723	1.13198	0.565990	−	0.824412i	$-0.308494\pi$
$773$	31.4723	1.13198	0.565990	+	0.824412i	$0.308494\pi$
$774$	0	0
$775$	−2.85169	−0.102436
$776$	0	0
$777$	−31.4492	−1.12823
$778$	0	0
$779$	−37.2805	−1.33571
$780$	0	0
$781$	68.1992	2.44036
$782$	0	0
$783$	46.1337	1.64868
$784$	0	0
$785$	24.6617	0.880215
$786$	0	0
$787$	34.7445	1.23851	0.619254	−	0.785191i	$-0.287435\pi$
$787$	34.7445	1.23851	0.619254	+	0.785191i	$0.287435\pi$
$788$	0	0
$789$	−52.0310	−1.85235
$790$	0	0
$791$	−21.1640	−0.752506
$792$	0	0
$793$	−0.339543	−0.0120575
$794$	0	0
$795$	45.6161	1.61783
$796$	0	0
$797$	−14.0976	−0.499361	−0.249681	−	0.968328i	$-0.580326\pi$
$797$	−14.0976	−0.499361	−0.249681	+	0.968328i	$0.580326\pi$
$798$	0	0
$799$	24.7801	0.876656
$800$	0	0
$801$	−36.7095	−1.29707
$802$	0	0
$803$	38.5328	1.35979
$804$	0	0
$805$	−36.7146	−1.29402
$806$	0	0
$807$	12.8458	0.452194
$808$	0	0
$809$	−35.5847	−1.25109	−0.625546	−	0.780187i	$-0.715124\pi$
$809$	−35.5847	−1.25109	−0.625546	+	0.780187i	$0.715124\pi$
$810$	0	0
$811$	−26.3501	−0.925276	−0.462638	−	0.886547i	$-0.653097\pi$
$811$	−26.3501	−0.925276	−0.462638	+	0.886547i	$0.653097\pi$
$812$	0	0
$813$	−54.5481	−1.91309
$814$	0	0
$815$	−6.89702	−0.241592
$816$	0	0
$817$	−14.6626	−0.512980
$818$	0	0
$819$	65.9981	2.30616
$820$	0	0
$821$	9.63192	0.336156	0.168078	−	0.985774i	$-0.446244\pi$
$821$	9.63192	0.336156	0.168078	+	0.985774i	$0.446244\pi$
$822$	0	0
$823$	−49.7000	−1.73243	−0.866216	−	0.499669i	$-0.833455\pi$
$823$	−49.7000	−1.73243	−0.866216	+	0.499669i	$0.833455\pi$
$824$	0	0
$825$	−66.4723	−2.31427
$826$	0	0
$827$	−20.8993	−0.726741	−0.363370	−	0.931645i	$-0.618374\pi$
$827$	−20.8993	−0.726741	−0.363370	+	0.931645i	$0.618374\pi$
$828$	0	0
$829$	8.63825	0.300019	0.150009	−	0.988685i	$-0.452070\pi$
$829$	8.63825	0.300019	0.150009	+	0.988685i	$0.452070\pi$
$830$	0	0
$831$	−65.7328	−2.28024
$832$	0	0
$833$	−140.163	−4.85637
$834$	0	0
$835$	3.70008	0.128047
$836$	0	0
$837$	4.99432	0.172629
$838$	0	0
$839$	23.7511	0.819979	0.409990	−	0.912090i	$-0.365532\pi$
$839$	23.7511	0.819979	0.409990	+	0.912090i	$0.365532\pi$
$840$	0	0
$841$	20.1116	0.693503
$842$	0	0
$843$	−71.3397	−2.45707
$844$	0	0
$845$	21.4451	0.737734
$846$	0	0
$847$	−139.155	−4.78142
$848$	0	0
$849$	18.4609	0.633577
$850$	0	0
$851$	5.00401	0.171535
$852$	0	0
$853$	1.48541	0.0508594	0.0254297	−	0.999677i	$-0.491905\pi$
$853$	1.48541	0.0508594	0.0254297	+	0.999677i	$0.491905\pi$
$854$	0	0
$855$	51.0094	1.74449
$856$	0	0
$857$	−35.5643	−1.21485	−0.607427	−	0.794375i	$-0.707798\pi$
$857$	−35.5643	−1.21485	−0.607427	+	0.794375i	$0.707798\pi$
$858$	0	0
$859$	−47.3663	−1.61612	−0.808058	−	0.589103i	$-0.799481\pi$
$859$	−47.3663	−1.61612	−0.808058	+	0.589103i	$0.799481\pi$
$860$	0	0
$861$	171.317	5.83846
$862$	0	0
$863$	21.7236	0.739481	0.369741	−	0.929135i	$-0.379447\pi$
$863$	21.7236	0.739481	0.369741	+	0.929135i	$0.379447\pi$
$864$	0	0
$865$	62.1793	2.11416
$866$	0	0
$867$	−91.2662	−3.09956
$868$	0	0
$869$	−59.7546	−2.02704
$870$	0	0
$871$	−23.4600	−0.794911
$872$	0	0
$873$	8.11010	0.274485
$874$	0	0
$875$	−19.1166	−0.646259
$876$	0	0
$877$	8.01629	0.270691	0.135345	−	0.990798i	$-0.456786\pi$
$877$	8.01629	0.270691	0.135345	+	0.990798i	$0.456786\pi$
$878$	0	0
$879$	−69.3013	−2.33747
$880$	0	0
$881$	4.36767	0.147151	0.0735753	−	0.997290i	$-0.476559\pi$
$881$	4.36767	0.147151	0.0735753	+	0.997290i	$0.476559\pi$
$882$	0	0
$883$	31.2357	1.05117	0.525583	−	0.850742i	$-0.323847\pi$
$883$	31.2357	1.05117	0.525583	+	0.850742i	$0.323847\pi$
$884$	0	0
$885$	18.9979	0.638607
$886$	0	0
$887$	9.31283	0.312694	0.156347	−	0.987702i	$-0.450028\pi$
$887$	9.31283	0.312694	0.156347	+	0.987702i	$0.450028\pi$
$888$	0	0
$889$	−54.1952	−1.81765
$890$	0	0
$891$	18.9825	0.635938
$892$	0	0
$893$	11.5758	0.387371
$894$	0	0
$895$	36.1591	1.20867
$896$	0	0
$897$	−16.4601	−0.549588
$898$	0	0
$899$	5.31670	0.177322
$900$	0	0
$901$	−37.3666	−1.24486
$902$	0	0
$903$	67.3797	2.24226
$904$	0	0
$905$	47.8380	1.59019
$906$	0	0
$907$	20.9680	0.696230	0.348115	−	0.937452i	$-0.386822\pi$
$907$	20.9680	0.696230	0.348115	+	0.937452i	$0.386822\pi$
$908$	0	0
$909$	−95.8973	−3.18071
$910$	0	0
$911$	−3.01219	−0.0997983	−0.0498991	−	0.998754i	$-0.515890\pi$
$911$	−3.01219	−0.0997983	−0.0498991	+	0.998754i	$0.515890\pi$
$912$	0	0
$913$	91.5998	3.03151
$914$	0	0
$915$	1.20596	0.0398677
$916$	0	0
$917$	2.61618	0.0863938
$918$	0	0
$919$	42.5684	1.40420	0.702100	−	0.712078i	$-0.252246\pi$
$919$	42.5684	1.40420	0.702100	+	0.712078i	$0.252246\pi$
$920$	0	0
$921$	−36.8879	−1.21550
$922$	0	0
$923$	−26.6296	−0.876526
$924$	0	0
$925$	−7.89063	−0.259442
$926$	0	0
$927$	89.8004	2.94943
$928$	0	0
$929$	21.6127	0.709090	0.354545	−	0.935039i	$-0.384636\pi$
$929$	21.6127	0.709090	0.354545	+	0.935039i	$0.384636\pi$
$930$	0	0
$931$	−65.4763	−2.14590
$932$	0	0
$933$	−55.0261	−1.80147
$934$	0	0
$935$	126.882	4.14949
$936$	0	0
$937$	−13.5508	−0.442684	−0.221342	−	0.975196i	$-0.571044\pi$
$937$	−13.5508	−0.442684	−0.221342	+	0.975196i	$0.571044\pi$
$938$	0	0
$939$	−28.5098	−0.930382
$940$	0	0
$941$	−10.5611	−0.344283	−0.172142	−	0.985072i	$-0.555069\pi$
$941$	−10.5611	−0.344283	−0.172142	+	0.985072i	$0.555069\pi$
$942$	0	0
$943$	−27.2589	−0.887671
$944$	0	0
$945$	−101.392	−3.29830
$946$	0	0
$947$	−1.14008	−0.0370478	−0.0185239	−	0.999828i	$-0.505897\pi$
$947$	−1.14008	−0.0370478	−0.0185239	+	0.999828i	$0.505897\pi$
$948$	0	0
$949$	−15.0459	−0.488410
$950$	0	0
$951$	−14.0051	−0.454146
$952$	0	0
$953$	−7.99572	−0.259007	−0.129503	−	0.991579i	$-0.541338\pi$
$953$	−7.99572	−0.259007	−0.129503	+	0.991579i	$0.541338\pi$
$954$	0	0
$955$	10.9316	0.353739
$956$	0	0
$957$	123.931	4.00612
$958$	0	0
$959$	100.779	3.25432
$960$	0	0
$961$	−30.4244	−0.981433
$962$	0	0
$963$	9.39557	0.302768
$964$	0	0
$965$	68.3760	2.20110
$966$	0	0
$967$	48.3744	1.55562	0.777808	−	0.628502i	$-0.216332\pi$
$967$	48.3744	1.55562	0.777808	+	0.628502i	$0.216332\pi$
$968$	0	0
$969$	−65.4953	−2.10401
$970$	0	0
$971$	13.5756	0.435663	0.217832	−	0.975986i	$-0.430102\pi$
$971$	13.5756	0.435663	0.217832	+	0.975986i	$0.430102\pi$
$972$	0	0
$973$	−44.7797	−1.43557
$974$	0	0
$975$	25.9554	0.831237
$976$	0	0
$977$	−4.43993	−0.142046	−0.0710230	−	0.997475i	$-0.522626\pi$
$977$	−4.43993	−0.142046	−0.0710230	+	0.997475i	$0.522626\pi$
$978$	0	0
$979$	−42.6558	−1.36328
$980$	0	0
$981$	−7.93763	−0.253429
$982$	0	0
$983$	0.923177	0.0294448	0.0147224	−	0.999892i	$-0.495314\pi$
$983$	0.923177	0.0294448	0.0147224	+	0.999892i	$0.495314\pi$
$984$	0	0
$985$	67.8907	2.16318
$986$	0	0
$987$	−53.1949	−1.69321
$988$	0	0
$989$	−10.7211	−0.340910
$990$	0	0
$991$	−12.4227	−0.394619	−0.197309	−	0.980341i	$-0.563220\pi$
$991$	−12.4227	−0.394619	−0.197309	+	0.980341i	$0.563220\pi$
$992$	0	0
$993$	63.4749	2.01432
$994$	0	0
$995$	34.8716	1.10550
$996$	0	0
$997$	−50.0650	−1.58557	−0.792787	−	0.609499i	$-0.791370\pi$
$997$	−50.0650	−1.58557	−0.792787	+	0.609499i	$0.791370\pi$
$998$	0	0
$999$	13.8193	0.437222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.c.1.3	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.c.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.87868 q^{3} -2.95953 q^{5} -5.20423 q^{7} +5.28682 q^{9} +O(q^{10})\)	\(q-2.87868 q^{3} -2.95953 q^{5} -5.20423 q^{7} +5.28682 q^{9} +6.14319 q^{11} -2.39873 q^{13} +8.51956 q^{15} -6.97884 q^{17} -3.26012 q^{19} +14.9813 q^{21} -2.38374 q^{23} +3.75883 q^{25} -6.58303 q^{27} -7.00797 q^{29} -0.758665 q^{31} -17.6843 q^{33} +15.4021 q^{35} -2.09923 q^{37} +6.90517 q^{39} +11.4353 q^{41} +4.49758 q^{43} -15.6465 q^{45} -3.55075 q^{47} +20.0840 q^{49} +20.0899 q^{51} +5.35427 q^{53} -18.1810 q^{55} +9.38484 q^{57} +2.22991 q^{59} +0.141552 q^{61} -27.5138 q^{63} +7.09910 q^{65} +9.78018 q^{67} +6.86204 q^{69} +11.1016 q^{71} +6.27245 q^{73} -10.8205 q^{75} -31.9706 q^{77} -9.72696 q^{79} +3.09001 q^{81} +14.9108 q^{83} +20.6541 q^{85} +20.1737 q^{87} -6.94358 q^{89} +12.4835 q^{91} +2.18396 q^{93} +9.64842 q^{95} +1.53402 q^{97} +32.4780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * q^99

Introduction

L-functions

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