LMFDB - Embedded newform 6008.2.a.d.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.55321 q^{3} -4.16598 q^{5} -2.08978 q^{7} +3.51888 q^{9} +O(q^{10})$	$q-2.55321 q^{3} -4.16598 q^{5} -2.08978 q^{7} +3.51888 q^{9} +1.07408 q^{11} +1.35848 q^{13} +10.6366 q^{15} +3.15848 q^{17} +1.85354 q^{19} +5.33566 q^{21} -3.48302 q^{23} +12.3554 q^{25} -1.32482 q^{27} -9.50723 q^{29} +8.19868 q^{31} -2.74235 q^{33} +8.70600 q^{35} -1.31894 q^{37} -3.46849 q^{39} -9.93589 q^{41} -8.76241 q^{43} -14.6596 q^{45} -10.8296 q^{47} -2.63280 q^{49} -8.06427 q^{51} -8.26073 q^{53} -4.47459 q^{55} -4.73248 q^{57} -3.05436 q^{59} -11.6854 q^{61} -7.35371 q^{63} -5.65941 q^{65} +9.10501 q^{67} +8.89289 q^{69} +6.64661 q^{71} +11.6562 q^{73} -31.5459 q^{75} -2.24459 q^{77} -3.49169 q^{79} -7.17411 q^{81} -12.0592 q^{83} -13.1582 q^{85} +24.2740 q^{87} -12.8712 q^{89} -2.83893 q^{91} -20.9330 q^{93} -7.72181 q^{95} +13.7435 q^{97} +3.77955 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.55321	−1.47410	−0.737048	−	0.675840i	$-0.763781\pi$
$3$	−2.55321	−1.47410	−0.737048	+	0.675840i	$0.763781\pi$
$4$	0	0
$5$	−4.16598	−1.86308	−0.931541	−	0.363635i	$-0.881535\pi$
$5$	−4.16598	−1.86308	−0.931541	+	0.363635i	$0.881535\pi$
$6$	0	0
$7$	−2.08978	−0.789864	−0.394932	−	0.918710i	$-0.629232\pi$
$7$	−2.08978	−0.789864	−0.394932	+	0.918710i	$0.629232\pi$
$8$	0	0
$9$	3.51888	1.17296
$10$	0	0
$11$	1.07408	0.323847	0.161923	−	0.986803i	$-0.448230\pi$
$11$	1.07408	0.323847	0.161923	+	0.986803i	$0.448230\pi$
$12$	0	0
$13$	1.35848	0.376775	0.188388	−	0.982095i	$-0.439674\pi$
$13$	1.35848	0.376775	0.188388	+	0.982095i	$0.439674\pi$
$14$	0	0
$15$	10.6366	2.74636
$16$	0	0
$17$	3.15848	0.766045	0.383022	−	0.923739i	$-0.374883\pi$
$17$	3.15848	0.766045	0.383022	+	0.923739i	$0.374883\pi$
$18$	0	0
$19$	1.85354	0.425231	0.212616	−	0.977136i	$-0.431802\pi$
$19$	1.85354	0.425231	0.212616	+	0.977136i	$0.431802\pi$
$20$	0	0
$21$	5.33566	1.16434
$22$	0	0
$23$	−3.48302	−0.726261	−0.363130	−	0.931738i	$-0.618292\pi$
$23$	−3.48302	−0.726261	−0.363130	+	0.931738i	$0.618292\pi$
$24$	0	0
$25$	12.3554	2.47108
$26$	0	0
$27$	−1.32482	−0.254961
$28$	0	0
$29$	−9.50723	−1.76545	−0.882724	−	0.469891i	$-0.844293\pi$
$29$	−9.50723	−1.76545	−0.882724	+	0.469891i	$0.844293\pi$
$30$	0	0
$31$	8.19868	1.47253	0.736263	−	0.676695i	$-0.236588\pi$
$31$	8.19868	1.47253	0.736263	+	0.676695i	$0.236588\pi$
$32$	0	0
$33$	−2.74235	−0.477381
$34$	0	0
$35$	8.70600	1.47158
$36$	0	0
$37$	−1.31894	−0.216832	−0.108416	−	0.994106i	$-0.534578\pi$
$37$	−1.31894	−0.216832	−0.108416	+	0.994106i	$0.534578\pi$
$38$	0	0
$39$	−3.46849	−0.555403
$40$	0	0
$41$	−9.93589	−1.55173	−0.775863	−	0.630902i	$-0.782685\pi$
$41$	−9.93589	−1.55173	−0.775863	+	0.630902i	$0.782685\pi$
$42$	0	0
$43$	−8.76241	−1.33625	−0.668127	−	0.744047i	$-0.732904\pi$
$43$	−8.76241	−1.33625	−0.668127	+	0.744047i	$0.732904\pi$
$44$	0	0
$45$	−14.6596	−2.18532
$46$	0	0
$47$	−10.8296	−1.57966	−0.789829	−	0.613327i	$-0.789831\pi$
$47$	−10.8296	−1.57966	−0.789829	+	0.613327i	$0.789831\pi$
$48$	0	0
$49$	−2.63280	−0.376114
$50$	0	0
$51$	−8.06427	−1.12922
$52$	0	0
$53$	−8.26073	−1.13470	−0.567349	−	0.823478i	$-0.692031\pi$
$53$	−8.26073	−1.13470	−0.567349	+	0.823478i	$0.692031\pi$
$54$	0	0
$55$	−4.47459	−0.603353
$56$	0	0
$57$	−4.73248	−0.626832
$58$	0	0
$59$	−3.05436	−0.397644	−0.198822	−	0.980036i	$-0.563712\pi$
$59$	−3.05436	−0.397644	−0.198822	+	0.980036i	$0.563712\pi$
$60$	0	0
$61$	−11.6854	−1.49616	−0.748078	−	0.663610i	$-0.769023\pi$
$61$	−11.6854	−1.49616	−0.748078	+	0.663610i	$0.769023\pi$
$62$	0	0
$63$	−7.35371	−0.926480
$64$	0	0
$65$	−5.65941	−0.701963
$66$	0	0
$67$	9.10501	1.11235	0.556177	−	0.831064i	$-0.312268\pi$
$67$	9.10501	1.11235	0.556177	+	0.831064i	$0.312268\pi$
$68$	0	0
$69$	8.89289	1.07058
$70$	0	0
$71$	6.64661	0.788808	0.394404	−	0.918937i	$-0.370951\pi$
$71$	6.64661	0.788808	0.394404	+	0.918937i	$0.370951\pi$
$72$	0	0
$73$	11.6562	1.36426	0.682129	−	0.731232i	$-0.261054\pi$
$73$	11.6562	1.36426	0.682129	+	0.731232i	$0.261054\pi$
$74$	0	0
$75$	−31.5459	−3.64261
$76$	0	0
$77$	−2.24459	−0.255795
$78$	0	0
$79$	−3.49169	−0.392846	−0.196423	−	0.980519i	$-0.562933\pi$
$79$	−3.49169	−0.392846	−0.196423	+	0.980519i	$0.562933\pi$
$80$	0	0
$81$	−7.17411	−0.797124
$82$	0	0
$83$	−12.0592	−1.32367	−0.661837	−	0.749648i	$-0.730223\pi$
$83$	−12.0592	−1.32367	−0.661837	+	0.749648i	$0.730223\pi$
$84$	0	0
$85$	−13.1582	−1.42721
$86$	0	0
$87$	24.2740	2.60244
$88$	0	0
$89$	−12.8712	−1.36434	−0.682170	−	0.731194i	$-0.738963\pi$
$89$	−12.8712	−1.36434	−0.682170	+	0.731194i	$0.738963\pi$
$90$	0	0
$91$	−2.83893	−0.297601
$92$	0	0
$93$	−20.9330	−2.17065
$94$	0	0
$95$	−7.72181	−0.792241
$96$	0	0
$97$	13.7435	1.39544	0.697719	−	0.716371i	$-0.254198\pi$
$97$	13.7435	1.39544	0.697719	+	0.716371i	$0.254198\pi$
$98$	0	0
$99$	3.77955	0.379859
$100$	0	0
$101$	4.57872	0.455600	0.227800	−	0.973708i	$-0.426847\pi$
$101$	4.57872	0.455600	0.227800	+	0.973708i	$0.426847\pi$
$102$	0	0
$103$	−3.51036	−0.345886	−0.172943	−	0.984932i	$-0.555328\pi$
$103$	−3.51036	−0.345886	−0.172943	+	0.984932i	$0.555328\pi$
$104$	0	0
$105$	−22.2282	−2.16925
$106$	0	0
$107$	12.4879	1.20725	0.603626	−	0.797267i	$-0.293722\pi$
$107$	12.4879	1.20725	0.603626	+	0.797267i	$0.293722\pi$
$108$	0	0
$109$	2.43852	0.233568	0.116784	−	0.993157i	$-0.462742\pi$
$109$	2.43852	0.233568	0.116784	+	0.993157i	$0.462742\pi$
$110$	0	0
$111$	3.36753	0.319632
$112$	0	0
$113$	11.1557	1.04944	0.524722	−	0.851274i	$-0.324169\pi$
$113$	11.1557	1.04944	0.524722	+	0.851274i	$0.324169\pi$
$114$	0	0
$115$	14.5102	1.35308
$116$	0	0
$117$	4.78034	0.441942
$118$	0	0
$119$	−6.60055	−0.605071
$120$	0	0
$121$	−9.84636	−0.895123
$122$	0	0
$123$	25.3684	2.28739
$124$	0	0
$125$	−30.6424	−2.74074
$126$	0	0
$127$	9.53184	0.845814	0.422907	−	0.906173i	$-0.361010\pi$
$127$	9.53184	0.845814	0.422907	+	0.906173i	$0.361010\pi$
$128$	0	0
$129$	22.3723	1.96977
$130$	0	0
$131$	20.0004	1.74744	0.873720	−	0.486428i	$-0.161701\pi$
$131$	20.0004	1.74744	0.873720	+	0.486428i	$0.161701\pi$
$132$	0	0
$133$	−3.87350	−0.335875
$134$	0	0
$135$	5.51915	0.475013
$136$	0	0
$137$	5.23097	0.446912	0.223456	−	0.974714i	$-0.428266\pi$
$137$	5.23097	0.446912	0.223456	+	0.974714i	$0.428266\pi$
$138$	0	0
$139$	−13.2093	−1.12040	−0.560200	−	0.828358i	$-0.689276\pi$
$139$	−13.2093	−1.12040	−0.560200	+	0.828358i	$0.689276\pi$
$140$	0	0
$141$	27.6502	2.32857
$142$	0	0
$143$	1.45912	0.122017
$144$	0	0
$145$	39.6069	3.28918
$146$	0	0
$147$	6.72209	0.554429
$148$	0	0
$149$	−4.36893	−0.357917	−0.178958	−	0.983857i	$-0.557273\pi$
$149$	−4.36893	−0.357917	−0.178958	+	0.983857i	$0.557273\pi$
$150$	0	0
$151$	−18.9958	−1.54586	−0.772928	−	0.634494i	$-0.781208\pi$
$151$	−18.9958	−1.54586	−0.772928	+	0.634494i	$0.781208\pi$
$152$	0	0
$153$	11.1143	0.898541
$154$	0	0
$155$	−34.1555	−2.74344
$156$	0	0
$157$	5.45222	0.435134	0.217567	−	0.976045i	$-0.430188\pi$
$157$	5.45222	0.435134	0.217567	+	0.976045i	$0.430188\pi$
$158$	0	0
$159$	21.0914	1.67265
$160$	0	0
$161$	7.27877	0.573647
$162$	0	0
$163$	−15.7359	−1.23253	−0.616266	−	0.787538i	$-0.711356\pi$
$163$	−15.7359	−1.23253	−0.616266	+	0.787538i	$0.711356\pi$
$164$	0	0
$165$	11.4246	0.889401
$166$	0	0
$167$	10.1779	0.787589	0.393795	−	0.919198i	$-0.371162\pi$
$167$	10.1779	0.787589	0.393795	+	0.919198i	$0.371162\pi$
$168$	0	0
$169$	−11.1545	−0.858041
$170$	0	0
$171$	6.52239	0.498780
$172$	0	0
$173$	−9.40791	−0.715270	−0.357635	−	0.933861i	$-0.616417\pi$
$173$	−9.40791	−0.715270	−0.357635	+	0.933861i	$0.616417\pi$
$174$	0	0
$175$	−25.8201	−1.95182
$176$	0	0
$177$	7.79843	0.586166
$178$	0	0
$179$	−15.6208	−1.16755	−0.583776	−	0.811915i	$-0.698425\pi$
$179$	−15.6208	−1.16755	−0.583776	+	0.811915i	$0.698425\pi$
$180$	0	0
$181$	−5.63934	−0.419169	−0.209584	−	0.977791i	$-0.567211\pi$
$181$	−5.63934	−0.419169	−0.209584	+	0.977791i	$0.567211\pi$
$182$	0	0
$183$	29.8352	2.20548
$184$	0	0
$185$	5.49467	0.403976
$186$	0	0
$187$	3.39246	0.248081
$188$	0	0
$189$	2.76858	0.201384
$190$	0	0
$191$	−6.51065	−0.471094	−0.235547	−	0.971863i	$-0.575688\pi$
$191$	−6.51065	−0.471094	−0.235547	+	0.971863i	$0.575688\pi$
$192$	0	0
$193$	−9.08195	−0.653733	−0.326867	−	0.945070i	$-0.605993\pi$
$193$	−9.08195	−0.653733	−0.326867	+	0.945070i	$0.605993\pi$
$194$	0	0
$195$	14.4497	1.03476
$196$	0	0
$197$	−22.1710	−1.57962	−0.789808	−	0.613355i	$-0.789820\pi$
$197$	−22.1710	−1.57962	−0.789808	+	0.613355i	$0.789820\pi$
$198$	0	0
$199$	22.3480	1.58421	0.792105	−	0.610384i	$-0.208985\pi$
$199$	22.3480	1.58421	0.792105	+	0.610384i	$0.208985\pi$
$200$	0	0
$201$	−23.2470	−1.63972
$202$	0	0
$203$	19.8681	1.39446
$204$	0	0
$205$	41.3927	2.89099
$206$	0	0
$207$	−12.2564	−0.851875
$208$	0	0
$209$	1.99085	0.137710
$210$	0	0
$211$	17.9515	1.23584	0.617918	−	0.786243i	$-0.287976\pi$
$211$	17.9515	1.23584	0.617918	+	0.786243i	$0.287976\pi$
$212$	0	0
$213$	−16.9702	−1.16278
$214$	0	0
$215$	36.5040	2.48955
$216$	0	0
$217$	−17.1335	−1.16310
$218$	0	0
$219$	−29.7608	−2.01105
$220$	0	0
$221$	4.29074	0.288627
$222$	0	0
$223$	−9.89954	−0.662922	−0.331461	−	0.943469i	$-0.607542\pi$
$223$	−9.89954	−0.662922	−0.331461	+	0.943469i	$0.607542\pi$
$224$	0	0
$225$	43.4772	2.89848
$226$	0	0
$227$	9.52507	0.632201	0.316100	−	0.948726i	$-0.397626\pi$
$227$	9.52507	0.632201	0.316100	+	0.948726i	$0.397626\pi$
$228$	0	0
$229$	−16.2093	−1.07114	−0.535570	−	0.844491i	$-0.679903\pi$
$229$	−16.2093	−1.07114	−0.535570	+	0.844491i	$0.679903\pi$
$230$	0	0
$231$	5.73091	0.377066
$232$	0	0
$233$	10.8645	0.711758	0.355879	−	0.934532i	$-0.384181\pi$
$233$	10.8645	0.711758	0.355879	+	0.934532i	$0.384181\pi$
$234$	0	0
$235$	45.1158	2.94303
$236$	0	0
$237$	8.91503	0.579093
$238$	0	0
$239$	−8.39814	−0.543230	−0.271615	−	0.962406i	$-0.587558\pi$
$239$	−8.39814	−0.543230	−0.271615	+	0.962406i	$0.587558\pi$
$240$	0	0
$241$	−8.97407	−0.578071	−0.289035	−	0.957318i	$-0.593335\pi$
$241$	−8.97407	−0.578071	−0.289035	+	0.957318i	$0.593335\pi$
$242$	0	0
$243$	22.2915	1.43000
$244$	0	0
$245$	10.9682	0.700732
$246$	0	0
$247$	2.51800	0.160217
$248$	0	0
$249$	30.7898	1.95122
$250$	0	0
$251$	−4.74339	−0.299400	−0.149700	−	0.988731i	$-0.547831\pi$
$251$	−4.74339	−0.299400	−0.149700	+	0.988731i	$0.547831\pi$
$252$	0	0
$253$	−3.74104	−0.235197
$254$	0	0
$255$	33.5956	2.10384
$256$	0	0
$257$	−29.6930	−1.85220	−0.926099	−	0.377279i	$-0.876860\pi$
$257$	−29.6930	−1.85220	−0.926099	+	0.377279i	$0.876860\pi$
$258$	0	0
$259$	2.75630	0.171268
$260$	0	0
$261$	−33.4548	−2.07080
$262$	0	0
$263$	28.0392	1.72897	0.864486	−	0.502657i	$-0.167644\pi$
$263$	28.0392	1.72897	0.864486	+	0.502657i	$0.167644\pi$
$264$	0	0
$265$	34.4140	2.11404
$266$	0	0
$267$	32.8628	2.01117
$268$	0	0
$269$	−19.5097	−1.18953	−0.594764	−	0.803900i	$-0.702755\pi$
$269$	−19.5097	−1.18953	−0.594764	+	0.803900i	$0.702755\pi$
$270$	0	0
$271$	17.0496	1.03569	0.517845	−	0.855474i	$-0.326734\pi$
$271$	17.0496	1.03569	0.517845	+	0.855474i	$0.326734\pi$
$272$	0	0
$273$	7.24840	0.438693
$274$	0	0
$275$	13.2706	0.800250
$276$	0	0
$277$	−16.1714	−0.971642	−0.485821	−	0.874058i	$-0.661479\pi$
$277$	−16.1714	−0.971642	−0.485821	+	0.874058i	$0.661479\pi$
$278$	0	0
$279$	28.8502	1.72722
$280$	0	0
$281$	3.95426	0.235891	0.117946	−	0.993020i	$-0.462369\pi$
$281$	3.95426	0.235891	0.117946	+	0.993020i	$0.462369\pi$
$282$	0	0
$283$	8.89590	0.528807	0.264403	−	0.964412i	$-0.414825\pi$
$283$	8.89590	0.528807	0.264403	+	0.964412i	$0.414825\pi$
$284$	0	0
$285$	19.7154	1.16784
$286$	0	0
$287$	20.7639	1.22565
$288$	0	0
$289$	−7.02398	−0.413175
$290$	0	0
$291$	−35.0900	−2.05701
$292$	0	0
$293$	−24.5702	−1.43541	−0.717703	−	0.696350i	$-0.754806\pi$
$293$	−24.5702	−1.43541	−0.717703	+	0.696350i	$0.754806\pi$
$294$	0	0
$295$	12.7244	0.740844
$296$	0	0
$297$	−1.42295	−0.0825682
$298$	0	0
$299$	−4.73163	−0.273637
$300$	0	0
$301$	18.3115	1.05546
$302$	0	0
$303$	−11.6904	−0.671598
$304$	0	0
$305$	48.6810	2.78746
$306$	0	0
$307$	−12.1743	−0.694821	−0.347411	−	0.937713i	$-0.612939\pi$
$307$	−12.1743	−0.694821	−0.347411	+	0.937713i	$0.612939\pi$
$308$	0	0
$309$	8.96268	0.509869
$310$	0	0
$311$	10.6890	0.606116	0.303058	−	0.952972i	$-0.401992\pi$
$311$	10.6890	0.606116	0.303058	+	0.952972i	$0.401992\pi$
$312$	0	0
$313$	6.30252	0.356239	0.178120	−	0.984009i	$-0.442999\pi$
$313$	6.30252	0.356239	0.178120	+	0.984009i	$0.442999\pi$
$314$	0	0
$315$	30.6354	1.72611
$316$	0	0
$317$	20.9677	1.17767	0.588833	−	0.808255i	$-0.299588\pi$
$317$	20.9677	1.17767	0.588833	+	0.808255i	$0.299588\pi$
$318$	0	0
$319$	−10.2115	−0.571735
$320$	0	0
$321$	−31.8843	−1.77961
$322$	0	0
$323$	5.85438	0.325746
$324$	0	0
$325$	16.7846	0.931040
$326$	0	0
$327$	−6.22605	−0.344301
$328$	0	0
$329$	22.6315	1.24772
$330$	0	0
$331$	−33.5565	−1.84443	−0.922217	−	0.386672i	$-0.873624\pi$
$331$	−33.5565	−1.84443	−0.922217	+	0.386672i	$0.873624\pi$
$332$	0	0
$333$	−4.64119	−0.254336
$334$	0	0
$335$	−37.9313	−2.07241
$336$	0	0
$337$	13.1558	0.716641	0.358321	−	0.933599i	$-0.383349\pi$
$337$	13.1558	0.716641	0.358321	+	0.933599i	$0.383349\pi$
$338$	0	0
$339$	−28.4830	−1.54698
$340$	0	0
$341$	8.80602	0.476873
$342$	0	0
$343$	20.1305	1.08694
$344$	0	0
$345$	−37.0476	−1.99458
$346$	0	0
$347$	16.2421	0.871924	0.435962	−	0.899965i	$-0.356408\pi$
$347$	16.2421	0.871924	0.435962	+	0.899965i	$0.356408\pi$
$348$	0	0
$349$	26.3248	1.40913	0.704567	−	0.709638i	$-0.251141\pi$
$349$	26.3248	1.40913	0.704567	+	0.709638i	$0.251141\pi$
$350$	0	0
$351$	−1.79974	−0.0960629
$352$	0	0
$353$	6.06170	0.322632	0.161316	−	0.986903i	$-0.448426\pi$
$353$	6.06170	0.322632	0.161316	+	0.986903i	$0.448426\pi$
$354$	0	0
$355$	−27.6897	−1.46961
$356$	0	0
$357$	16.8526	0.891934
$358$	0	0
$359$	−8.09862	−0.427429	−0.213714	−	0.976896i	$-0.568556\pi$
$359$	−8.09862	−0.427429	−0.213714	+	0.976896i	$0.568556\pi$
$360$	0	0
$361$	−15.5644	−0.819178
$362$	0	0
$363$	25.1398	1.31950
$364$	0	0
$365$	−48.5596	−2.54173
$366$	0	0
$367$	−8.82262	−0.460537	−0.230269	−	0.973127i	$-0.573960\pi$
$367$	−8.82262	−0.460537	−0.230269	+	0.973127i	$0.573960\pi$
$368$	0	0
$369$	−34.9632	−1.82011
$370$	0	0
$371$	17.2631	0.896257
$372$	0	0
$373$	−1.88554	−0.0976294	−0.0488147	−	0.998808i	$-0.515544\pi$
$373$	−1.88554	−0.0976294	−0.0488147	+	0.998808i	$0.515544\pi$
$374$	0	0
$375$	78.2365	4.04011
$376$	0	0
$377$	−12.9154	−0.665177
$378$	0	0
$379$	5.66929	0.291212	0.145606	−	0.989343i	$-0.453487\pi$
$379$	5.66929	0.291212	0.145606	+	0.989343i	$0.453487\pi$
$380$	0	0
$381$	−24.3368	−1.24681
$382$	0	0
$383$	33.5291	1.71326	0.856628	−	0.515934i	$-0.172555\pi$
$383$	33.5291	1.71326	0.856628	+	0.515934i	$0.172555\pi$
$384$	0	0
$385$	9.35092	0.476567
$386$	0	0
$387$	−30.8339	−1.56737
$388$	0	0
$389$	−34.3991	−1.74410	−0.872052	−	0.489412i	$-0.837211\pi$
$389$	−34.3991	−1.74410	−0.872052	+	0.489412i	$0.837211\pi$
$390$	0	0
$391$	−11.0011	−0.556348
$392$	0	0
$393$	−51.0652	−2.57590
$394$	0	0
$395$	14.5463	0.731905
$396$	0	0
$397$	−25.2104	−1.26527	−0.632636	−	0.774450i	$-0.718027\pi$
$397$	−25.2104	−1.26527	−0.632636	+	0.774450i	$0.718027\pi$
$398$	0	0
$399$	9.88986	0.495112
$400$	0	0
$401$	−16.7054	−0.834229	−0.417115	−	0.908854i	$-0.636959\pi$
$401$	−16.7054	−0.834229	−0.417115	+	0.908854i	$0.636959\pi$
$402$	0	0
$403$	11.1378	0.554811
$404$	0	0
$405$	29.8872	1.48511
$406$	0	0
$407$	−1.41664	−0.0702204
$408$	0	0
$409$	25.5961	1.26565	0.632824	−	0.774296i	$-0.281896\pi$
$409$	25.5961	1.26565	0.632824	+	0.774296i	$0.281896\pi$
$410$	0	0
$411$	−13.3558	−0.658792
$412$	0	0
$413$	6.38296	0.314085
$414$	0	0
$415$	50.2385	2.46611
$416$	0	0
$417$	33.7262	1.65158
$418$	0	0
$419$	19.5663	0.955877	0.477939	−	0.878393i	$-0.341384\pi$
$419$	19.5663	0.955877	0.477939	+	0.878393i	$0.341384\pi$
$420$	0	0
$421$	−3.59674	−0.175295	−0.0876473	−	0.996152i	$-0.527935\pi$
$421$	−3.59674	−0.175295	−0.0876473	+	0.996152i	$0.527935\pi$
$422$	0	0
$423$	−38.1080	−1.85288
$424$	0	0
$425$	39.0243	1.89296
$426$	0	0
$427$	24.4199	1.18176
$428$	0	0
$429$	−3.72543	−0.179865
$430$	0	0
$431$	−0.733251	−0.0353195	−0.0176597	−	0.999844i	$-0.505622\pi$
$431$	−0.733251	−0.0353195	−0.0176597	+	0.999844i	$0.505622\pi$
$432$	0	0
$433$	26.6968	1.28297	0.641483	−	0.767137i	$-0.278319\pi$
$433$	26.6968	1.28297	0.641483	+	0.767137i	$0.278319\pi$
$434$	0	0
$435$	−101.125	−4.84857
$436$	0	0
$437$	−6.45593	−0.308829
$438$	0	0
$439$	−31.1514	−1.48678	−0.743388	−	0.668861i	$-0.766782\pi$
$439$	−31.1514	−1.48678	−0.743388	+	0.668861i	$0.766782\pi$
$440$	0	0
$441$	−9.26452	−0.441167
$442$	0	0
$443$	8.85504	0.420716	0.210358	−	0.977624i	$-0.432537\pi$
$443$	8.85504	0.420716	0.210358	+	0.977624i	$0.432537\pi$
$444$	0	0
$445$	53.6210	2.54188
$446$	0	0
$447$	11.1548	0.527604
$448$	0	0
$449$	−9.27387	−0.437661	−0.218831	−	0.975763i	$-0.570224\pi$
$449$	−9.27387	−0.437661	−0.218831	+	0.975763i	$0.570224\pi$
$450$	0	0
$451$	−10.6719	−0.502521
$452$	0	0
$453$	48.5002	2.27874
$454$	0	0
$455$	11.8269	0.554456
$456$	0	0
$457$	−22.8567	−1.06919	−0.534595	−	0.845108i	$-0.679536\pi$
$457$	−22.8567	−1.06919	−0.534595	+	0.845108i	$0.679536\pi$
$458$	0	0
$459$	−4.18441	−0.195311
$460$	0	0
$461$	32.5438	1.51571	0.757857	−	0.652420i	$-0.226246\pi$
$461$	32.5438	1.51571	0.757857	+	0.652420i	$0.226246\pi$
$462$	0	0
$463$	−8.34987	−0.388051	−0.194026	−	0.980996i	$-0.562154\pi$
$463$	−8.34987	−0.388051	−0.194026	+	0.980996i	$0.562154\pi$
$464$	0	0
$465$	87.2063	4.04409
$466$	0	0
$467$	35.8937	1.66096	0.830482	−	0.557045i	$-0.188065\pi$
$467$	35.8937	1.66096	0.830482	+	0.557045i	$0.188065\pi$
$468$	0	0
$469$	−19.0275	−0.878609
$470$	0	0
$471$	−13.9207	−0.641430
$472$	0	0
$473$	−9.41151	−0.432742
$474$	0	0
$475$	22.9012	1.05078
$476$	0	0
$477$	−29.0685	−1.33096
$478$	0	0
$479$	9.50407	0.434252	0.217126	−	0.976144i	$-0.430332\pi$
$479$	9.50407	0.434252	0.217126	+	0.976144i	$0.430332\pi$
$480$	0	0
$481$	−1.79175	−0.0816970
$482$	0	0
$483$	−18.5842	−0.845612
$484$	0	0
$485$	−57.2550	−2.59982
$486$	0	0
$487$	−19.8984	−0.901680	−0.450840	−	0.892605i	$-0.648876\pi$
$487$	−19.8984	−0.901680	−0.450840	+	0.892605i	$0.648876\pi$
$488$	0	0
$489$	40.1771	1.81687
$490$	0	0
$491$	−21.2459	−0.958816	−0.479408	−	0.877592i	$-0.659149\pi$
$491$	−21.2459	−0.958816	−0.479408	+	0.877592i	$0.659149\pi$
$492$	0	0
$493$	−30.0284	−1.35241
$494$	0	0
$495$	−15.7455	−0.707710
$496$	0	0
$497$	−13.8900	−0.623051
$498$	0	0
$499$	4.78214	0.214078	0.107039	−	0.994255i	$-0.465863\pi$
$499$	4.78214	0.214078	0.107039	+	0.994255i	$0.465863\pi$
$500$	0	0
$501$	−25.9863	−1.16098
$502$	0	0
$503$	8.80584	0.392633	0.196316	−	0.980541i	$-0.437102\pi$
$503$	8.80584	0.392633	0.196316	+	0.980541i	$0.437102\pi$
$504$	0	0
$505$	−19.0749	−0.848820
$506$	0	0
$507$	28.4799	1.26483
$508$	0	0
$509$	13.0546	0.578636	0.289318	−	0.957233i	$-0.406571\pi$
$509$	13.0546	0.578636	0.289318	+	0.957233i	$0.406571\pi$
$510$	0	0
$511$	−24.3590	−1.07758
$512$	0	0
$513$	−2.45560	−0.108417
$514$	0	0
$515$	14.6241	0.644414
$516$	0	0
$517$	−11.6318	−0.511567
$518$	0	0
$519$	24.0204	1.05438
$520$	0	0
$521$	23.6570	1.03643	0.518215	−	0.855250i	$-0.326597\pi$
$521$	23.6570	1.03643	0.518215	+	0.855250i	$0.326597\pi$
$522$	0	0
$523$	34.1034	1.49124	0.745619	−	0.666372i	$-0.232154\pi$
$523$	34.1034	1.49124	0.745619	+	0.666372i	$0.232154\pi$
$524$	0	0
$525$	65.9241	2.87717
$526$	0	0
$527$	25.8954	1.12802
$528$	0	0
$529$	−10.8685	−0.472545
$530$	0	0
$531$	−10.7479	−0.466421
$532$	0	0
$533$	−13.4977	−0.584652
$534$	0	0
$535$	−52.0244	−2.24921
$536$	0	0
$537$	39.8831	1.72108
$538$	0	0
$539$	−2.82783	−0.121803
$540$	0	0
$541$	4.22659	0.181715	0.0908576	−	0.995864i	$-0.471039\pi$
$541$	4.22659	0.181715	0.0908576	+	0.995864i	$0.471039\pi$
$542$	0	0
$543$	14.3984	0.617895
$544$	0	0
$545$	−10.1588	−0.435156
$546$	0	0
$547$	20.7974	0.889232	0.444616	−	0.895721i	$-0.353340\pi$
$547$	20.7974	0.889232	0.444616	+	0.895721i	$0.353340\pi$
$548$	0	0
$549$	−41.1194	−1.75493
$550$	0	0
$551$	−17.6220	−0.750724
$552$	0	0
$553$	7.29689	0.310295
$554$	0	0
$555$	−14.0291	−0.595500
$556$	0	0
$557$	−45.3021	−1.91951	−0.959757	−	0.280832i	$-0.909389\pi$
$557$	−45.3021	−1.91951	−0.959757	+	0.280832i	$0.909389\pi$
$558$	0	0
$559$	−11.9036	−0.503467
$560$	0	0
$561$	−8.66166	−0.365695
$562$	0	0
$563$	−39.5246	−1.66576	−0.832881	−	0.553452i	$-0.813310\pi$
$563$	−39.5246	−1.66576	−0.832881	+	0.553452i	$0.813310\pi$
$564$	0	0
$565$	−46.4746	−1.95520
$566$	0	0
$567$	14.9924	0.629620
$568$	0	0
$569$	12.9707	0.543761	0.271880	−	0.962331i	$-0.412354\pi$
$569$	12.9707	0.543761	0.271880	+	0.962331i	$0.412354\pi$
$570$	0	0
$571$	17.9200	0.749927	0.374963	−	0.927040i	$-0.377655\pi$
$571$	17.9200	0.749927	0.374963	+	0.927040i	$0.377655\pi$
$572$	0	0
$573$	16.6231	0.694438
$574$	0	0
$575$	−43.0341	−1.79465
$576$	0	0
$577$	42.9733	1.78900	0.894500	−	0.447068i	$-0.147532\pi$
$577$	42.9733	1.78900	0.894500	+	0.447068i	$0.147532\pi$
$578$	0	0
$579$	23.1881	0.963666
$580$	0	0
$581$	25.2012	1.04552
$582$	0	0
$583$	−8.87266	−0.367468
$584$	0	0
$585$	−19.9148	−0.823375
$586$	0	0
$587$	3.64314	0.150368	0.0751841	−	0.997170i	$-0.476046\pi$
$587$	3.64314	0.150368	0.0751841	+	0.997170i	$0.476046\pi$
$588$	0	0
$589$	15.1966	0.626165
$590$	0	0
$591$	56.6071	2.32851
$592$	0	0
$593$	39.0738	1.60457	0.802284	−	0.596942i	$-0.203618\pi$
$593$	39.0738	1.60457	0.802284	+	0.596942i	$0.203618\pi$
$594$	0	0
$595$	27.4978	1.12730
$596$	0	0
$597$	−57.0593	−2.33528
$598$	0	0
$599$	−7.82266	−0.319625	−0.159813	−	0.987147i	$-0.551089\pi$
$599$	−7.82266	−0.319625	−0.159813	+	0.987147i	$0.551089\pi$
$600$	0	0
$601$	−26.3695	−1.07563	−0.537816	−	0.843062i	$-0.680751\pi$
$601$	−26.3695	−1.07563	−0.537816	+	0.843062i	$0.680751\pi$
$602$	0	0
$603$	32.0395	1.30475
$604$	0	0
$605$	41.0197	1.66769
$606$	0	0
$607$	36.0117	1.46167	0.730835	−	0.682554i	$-0.239131\pi$
$607$	36.0117	1.46167	0.730835	+	0.682554i	$0.239131\pi$
$608$	0	0
$609$	−50.7274	−2.05558
$610$	0	0
$611$	−14.7118	−0.595176
$612$	0	0
$613$	−23.6535	−0.955354	−0.477677	−	0.878535i	$-0.658521\pi$
$613$	−23.6535	−0.955354	−0.477677	+	0.878535i	$0.658521\pi$
$614$	0	0
$615$	−105.684	−4.26160
$616$	0	0
$617$	−8.93520	−0.359718	−0.179859	−	0.983692i	$-0.557564\pi$
$617$	−8.93520	−0.359718	−0.179859	+	0.983692i	$0.557564\pi$
$618$	0	0
$619$	43.0059	1.72855	0.864277	−	0.503016i	$-0.167776\pi$
$619$	43.0059	1.72855	0.864277	+	0.503016i	$0.167776\pi$
$620$	0	0
$621$	4.61436	0.185168
$622$	0	0
$623$	26.8979	1.07764
$624$	0	0
$625$	65.8787	2.63515
$626$	0	0
$627$	−5.08305	−0.202998
$628$	0	0
$629$	−4.16585	−0.166103
$630$	0	0
$631$	−18.0635	−0.719095	−0.359547	−	0.933127i	$-0.617069\pi$
$631$	−18.0635	−0.719095	−0.359547	+	0.933127i	$0.617069\pi$
$632$	0	0
$633$	−45.8341	−1.82174
$634$	0	0
$635$	−39.7094	−1.57582
$636$	0	0
$637$	−3.57661	−0.141711
$638$	0	0
$639$	23.3886	0.925241
$640$	0	0
$641$	−15.4254	−0.609267	−0.304633	−	0.952470i	$-0.598534\pi$
$641$	−15.4254	−0.609267	−0.304633	+	0.952470i	$0.598534\pi$
$642$	0	0
$643$	8.47957	0.334401	0.167201	−	0.985923i	$-0.446527\pi$
$643$	8.47957	0.334401	0.167201	+	0.985923i	$0.446527\pi$
$644$	0	0
$645$	−93.2024	−3.66984
$646$	0	0
$647$	−3.93454	−0.154683	−0.0773414	−	0.997005i	$-0.524643\pi$
$647$	−3.93454	−0.154683	−0.0773414	+	0.997005i	$0.524643\pi$
$648$	0	0
$649$	−3.28062	−0.128776
$650$	0	0
$651$	43.7454	1.71452
$652$	0	0
$653$	−25.3151	−0.990655	−0.495327	−	0.868706i	$-0.664952\pi$
$653$	−25.3151	−0.990655	−0.495327	+	0.868706i	$0.664952\pi$
$654$	0	0
$655$	−83.3212	−3.25563
$656$	0	0
$657$	41.0169	1.60022
$658$	0	0
$659$	11.2975	0.440086	0.220043	−	0.975490i	$-0.429380\pi$
$659$	11.2975	0.440086	0.220043	+	0.975490i	$0.429380\pi$
$660$	0	0
$661$	33.9309	1.31976	0.659879	−	0.751372i	$-0.270607\pi$
$661$	33.9309	1.31976	0.659879	+	0.751372i	$0.270607\pi$
$662$	0	0
$663$	−10.9552	−0.425464
$664$	0	0
$665$	16.1369	0.625763
$666$	0	0
$667$	33.1139	1.28218
$668$	0	0
$669$	25.2756	0.977211
$670$	0	0
$671$	−12.5510	−0.484525
$672$	0	0
$673$	12.7344	0.490876	0.245438	−	0.969412i	$-0.421068\pi$
$673$	12.7344	0.490876	0.245438	+	0.969412i	$0.421068\pi$
$674$	0	0
$675$	−16.3686	−0.630028
$676$	0	0
$677$	50.7250	1.94952	0.974759	−	0.223259i	$-0.0716696\pi$
$677$	50.7250	1.94952	0.974759	+	0.223259i	$0.0716696\pi$
$678$	0	0
$679$	−28.7209	−1.10221
$680$	0	0
$681$	−24.3195	−0.931925
$682$	0	0
$683$	−46.6930	−1.78666	−0.893329	−	0.449404i	$-0.851636\pi$
$683$	−46.6930	−1.78666	−0.893329	+	0.449404i	$0.851636\pi$
$684$	0	0
$685$	−21.7921	−0.832634
$686$	0	0
$687$	41.3857	1.57896
$688$	0	0
$689$	−11.2220	−0.427526
$690$	0	0
$691$	26.6783	1.01489	0.507445	−	0.861684i	$-0.330590\pi$
$691$	26.6783	1.01489	0.507445	+	0.861684i	$0.330590\pi$
$692$	0	0
$693$	−7.89845	−0.300037
$694$	0	0
$695$	55.0297	2.08740
$696$	0	0
$697$	−31.3824	−1.18869
$698$	0	0
$699$	−27.7394	−1.04920
$700$	0	0
$701$	26.5678	1.00345	0.501726	−	0.865027i	$-0.332699\pi$
$701$	26.5678	1.00345	0.501726	+	0.865027i	$0.332699\pi$
$702$	0	0
$703$	−2.44471	−0.0922039
$704$	0	0
$705$	−115.190	−4.33832
$706$	0	0
$707$	−9.56854	−0.359862
$708$	0	0
$709$	38.6904	1.45305	0.726525	−	0.687140i	$-0.241134\pi$
$709$	38.6904	1.45305	0.726525	+	0.687140i	$0.241134\pi$
$710$	0	0
$711$	−12.2869	−0.460793
$712$	0	0
$713$	−28.5562	−1.06944
$714$	0	0
$715$	−6.07864	−0.227328
$716$	0	0
$717$	21.4422	0.800774
$718$	0	0
$719$	0.593023	0.0221160	0.0110580	−	0.999939i	$-0.496480\pi$
$719$	0.593023	0.0221160	0.0110580	+	0.999939i	$0.496480\pi$
$720$	0	0
$721$	7.33589	0.273203
$722$	0	0
$723$	22.9127	0.852132
$724$	0	0
$725$	−117.466	−4.36256
$726$	0	0
$727$	−1.08439	−0.0402180	−0.0201090	−	0.999798i	$-0.506401\pi$
$727$	−1.08439	−0.0402180	−0.0201090	+	0.999798i	$0.506401\pi$
$728$	0	0
$729$	−35.3925	−1.31083
$730$	0	0
$731$	−27.6759	−1.02363
$732$	0	0
$733$	3.42005	0.126322	0.0631612	−	0.998003i	$-0.479882\pi$
$733$	3.42005	0.126322	0.0631612	+	0.998003i	$0.479882\pi$
$734$	0	0
$735$	−28.0041	−1.03295
$736$	0	0
$737$	9.77949	0.360232
$738$	0	0
$739$	30.2110	1.11133	0.555664	−	0.831407i	$-0.312464\pi$
$739$	30.2110	1.11133	0.555664	+	0.831407i	$0.312464\pi$
$740$	0	0
$741$	−6.42899	−0.236175
$742$	0	0
$743$	23.8230	0.873981	0.436991	−	0.899466i	$-0.356044\pi$
$743$	23.8230	0.873981	0.436991	+	0.899466i	$0.356044\pi$
$744$	0	0
$745$	18.2009	0.666829
$746$	0	0
$747$	−42.4350	−1.55262
$748$	0	0
$749$	−26.0971	−0.953566
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	12.1109	0.441344
$754$	0	0
$755$	79.1361	2.88006
$756$	0	0
$757$	16.2411	0.590292	0.295146	−	0.955452i	$-0.404632\pi$
$757$	16.2411	0.590292	0.295146	+	0.955452i	$0.404632\pi$
$758$	0	0
$759$	9.55166	0.346703
$760$	0	0
$761$	−7.12329	−0.258219	−0.129109	−	0.991630i	$-0.541212\pi$
$761$	−7.12329	−0.258219	−0.129109	+	0.991630i	$0.541212\pi$
$762$	0	0
$763$	−5.09598	−0.184487
$764$	0	0
$765$	−46.3021	−1.67406
$766$	0	0
$767$	−4.14930	−0.149822
$768$	0	0
$769$	26.4347	0.953260	0.476630	−	0.879104i	$-0.341858\pi$
$769$	26.4347	0.953260	0.476630	+	0.879104i	$0.341858\pi$
$770$	0	0
$771$	75.8125	2.73032
$772$	0	0
$773$	−0.816032	−0.0293506	−0.0146753	−	0.999892i	$-0.504671\pi$
$773$	−0.816032	−0.0293506	−0.0146753	+	0.999892i	$0.504671\pi$
$774$	0	0
$775$	101.298	3.63873
$776$	0	0
$777$	−7.03741	−0.252466
$778$	0	0
$779$	−18.4166	−0.659843
$780$	0	0
$781$	7.13898	0.255453
$782$	0	0
$783$	12.5953	0.450120
$784$	0	0
$785$	−22.7138	−0.810692
$786$	0	0
$787$	−22.2504	−0.793140	−0.396570	−	0.918004i	$-0.629800\pi$
$787$	−22.2504	−0.793140	−0.396570	+	0.918004i	$0.629800\pi$
$788$	0	0
$789$	−71.5900	−2.54867
$790$	0	0
$791$	−23.3131	−0.828918
$792$	0	0
$793$	−15.8743	−0.563715
$794$	0	0
$795$	−87.8662	−3.11629
$796$	0	0
$797$	19.6730	0.696855	0.348427	−	0.937336i	$-0.386716\pi$
$797$	19.6730	0.696855	0.348427	+	0.937336i	$0.386716\pi$
$798$	0	0
$799$	−34.2051	−1.21009
$800$	0	0
$801$	−45.2921	−1.60032
$802$	0	0
$803$	12.5197	0.441810
$804$	0	0
$805$	−30.3232	−1.06875
$806$	0	0
$807$	49.8124	1.75348
$808$	0	0
$809$	17.3975	0.611662	0.305831	−	0.952086i	$-0.401066\pi$
$809$	17.3975	0.611662	0.305831	+	0.952086i	$0.401066\pi$
$810$	0	0
$811$	−10.8849	−0.382220	−0.191110	−	0.981569i	$-0.561209\pi$
$811$	−10.8849	−0.382220	−0.191110	+	0.981569i	$0.561209\pi$
$812$	0	0
$813$	−43.5312	−1.52671
$814$	0	0
$815$	65.5556	2.29631
$816$	0	0
$817$	−16.2415	−0.568217
$818$	0	0
$819$	−9.98988	−0.349074
$820$	0	0
$821$	13.7780	0.480857	0.240428	−	0.970667i	$-0.422712\pi$
$821$	13.7780	0.480857	0.240428	+	0.970667i	$0.422712\pi$
$822$	0	0
$823$	46.3485	1.61561	0.807804	−	0.589451i	$-0.200656\pi$
$823$	46.3485	1.61561	0.807804	+	0.589451i	$0.200656\pi$
$824$	0	0
$825$	−33.8828	−1.17965
$826$	0	0
$827$	−15.7200	−0.546638	−0.273319	−	0.961923i	$-0.588121\pi$
$827$	−15.7200	−0.546638	−0.273319	+	0.961923i	$0.588121\pi$
$828$	0	0
$829$	−13.7758	−0.478454	−0.239227	−	0.970964i	$-0.576894\pi$
$829$	−13.7758	−0.478454	−0.239227	+	0.970964i	$0.576894\pi$
$830$	0	0
$831$	41.2889	1.43229
$832$	0	0
$833$	−8.31566	−0.288121
$834$	0	0
$835$	−42.4009	−1.46734
$836$	0	0
$837$	−10.8617	−0.375437
$838$	0	0
$839$	−27.5887	−0.952468	−0.476234	−	0.879319i	$-0.657998\pi$
$839$	−27.5887	−0.952468	−0.476234	+	0.879319i	$0.657998\pi$
$840$	0	0
$841$	61.3875	2.11681
$842$	0	0
$843$	−10.0960	−0.347726
$844$	0	0
$845$	46.4695	1.59860
$846$	0	0
$847$	20.5768	0.707026
$848$	0	0
$849$	−22.7131	−0.779512
$850$	0	0
$851$	4.59390	0.157477
$852$	0	0
$853$	−7.95243	−0.272286	−0.136143	−	0.990689i	$-0.543471\pi$
$853$	−7.95243	−0.272286	−0.136143	+	0.990689i	$0.543471\pi$
$854$	0	0
$855$	−27.1722	−0.929268
$856$	0	0
$857$	−24.9926	−0.853731	−0.426866	−	0.904315i	$-0.640382\pi$
$857$	−24.9926	−0.853731	−0.426866	+	0.904315i	$0.640382\pi$
$858$	0	0
$859$	41.2643	1.40792	0.703960	−	0.710239i	$-0.251413\pi$
$859$	41.2643	1.40792	0.703960	+	0.710239i	$0.251413\pi$
$860$	0	0
$861$	−53.0145	−1.80673
$862$	0	0
$863$	27.4960	0.935975	0.467988	−	0.883735i	$-0.344979\pi$
$863$	27.4960	0.935975	0.467988	+	0.883735i	$0.344979\pi$
$864$	0	0
$865$	39.1932	1.33261
$866$	0	0
$867$	17.9337	0.609060
$868$	0	0
$869$	−3.75035	−0.127222
$870$	0	0
$871$	12.3690	0.419107
$872$	0	0
$873$	48.3617	1.63679
$874$	0	0
$875$	64.0360	2.16481
$876$	0	0
$877$	29.6184	1.00014	0.500071	−	0.865984i	$-0.333307\pi$
$877$	29.6184	1.00014	0.500071	+	0.865984i	$0.333307\pi$
$878$	0	0
$879$	62.7328	2.11593
$880$	0	0
$881$	26.0156	0.876487	0.438243	−	0.898856i	$-0.355601\pi$
$881$	26.0156	0.876487	0.438243	+	0.898856i	$0.355601\pi$
$882$	0	0
$883$	−29.8169	−1.00342	−0.501709	−	0.865037i	$-0.667295\pi$
$883$	−29.8169	−1.00342	−0.501709	+	0.865037i	$0.667295\pi$
$884$	0	0
$885$	−32.4881	−1.09208
$886$	0	0
$887$	−0.570708	−0.0191625	−0.00958125	−	0.999954i	$-0.503050\pi$
$887$	−0.570708	−0.0191625	−0.00958125	+	0.999954i	$0.503050\pi$
$888$	0	0
$889$	−19.9195	−0.668078
$890$	0	0
$891$	−7.70556	−0.258146
$892$	0	0
$893$	−20.0731	−0.671720
$894$	0	0
$895$	65.0758	2.17525
$896$	0	0
$897$	12.0808	0.403367
$898$	0	0
$899$	−77.9468	−2.59967
$900$	0	0
$901$	−26.0914	−0.869229
$902$	0	0
$903$	−46.7532	−1.55585
$904$	0	0
$905$	23.4934	0.780946
$906$	0	0
$907$	−36.0219	−1.19609	−0.598044	−	0.801463i	$-0.704055\pi$
$907$	−36.0219	−1.19609	−0.598044	+	0.801463i	$0.704055\pi$
$908$	0	0
$909$	16.1120	0.534401
$910$	0	0
$911$	24.2596	0.803758	0.401879	−	0.915693i	$-0.368357\pi$
$911$	24.2596	0.803758	0.401879	+	0.915693i	$0.368357\pi$
$912$	0	0
$913$	−12.9526	−0.428667
$914$	0	0
$915$	−124.293	−4.10899
$916$	0	0
$917$	−41.7965	−1.38024
$918$	0	0
$919$	29.5445	0.974583	0.487291	−	0.873239i	$-0.337985\pi$
$919$	29.5445	0.974583	0.487291	+	0.873239i	$0.337985\pi$
$920$	0	0
$921$	31.0834	1.02423
$922$	0	0
$923$	9.02930	0.297203
$924$	0	0
$925$	−16.2960	−0.535809
$926$	0	0
$927$	−12.3525	−0.405710
$928$	0	0
$929$	−16.5070	−0.541578	−0.270789	−	0.962639i	$-0.587285\pi$
$929$	−16.5070	−0.541578	−0.270789	+	0.962639i	$0.587285\pi$
$930$	0	0
$931$	−4.88000	−0.159936
$932$	0	0
$933$	−27.2912	−0.893474
$934$	0	0
$935$	−14.1329	−0.462196
$936$	0	0
$937$	13.9256	0.454929	0.227464	−	0.973786i	$-0.426957\pi$
$937$	13.9256	0.454929	0.227464	+	0.973786i	$0.426957\pi$
$938$	0	0
$939$	−16.0917	−0.525131
$940$	0	0
$941$	−16.8957	−0.550784	−0.275392	−	0.961332i	$-0.588808\pi$
$941$	−16.8957	−0.550784	−0.275392	+	0.961332i	$0.588808\pi$
$942$	0	0
$943$	34.6070	1.12696
$944$	0	0
$945$	−11.5338	−0.375196
$946$	0	0
$947$	1.54343	0.0501548	0.0250774	−	0.999686i	$-0.492017\pi$
$947$	1.54343	0.0501548	0.0250774	+	0.999686i	$0.492017\pi$
$948$	0	0
$949$	15.8348	0.514019
$950$	0	0
$951$	−53.5350	−1.73599
$952$	0	0
$953$	−28.1365	−0.911429	−0.455715	−	0.890126i	$-0.650616\pi$
$953$	−28.1365	−0.911429	−0.455715	+	0.890126i	$0.650616\pi$
$954$	0	0
$955$	27.1232	0.877687
$956$	0	0
$957$	26.0721	0.842792
$958$	0	0
$959$	−10.9316	−0.353000
$960$	0	0
$961$	36.2184	1.16834
$962$	0	0
$963$	43.9435	1.41606
$964$	0	0
$965$	37.8352	1.21796
$966$	0	0
$967$	35.8753	1.15367	0.576836	−	0.816860i	$-0.304287\pi$
$967$	35.8753	1.15367	0.576836	+	0.816860i	$0.304287\pi$
$968$	0	0
$969$	−14.9475	−0.480182
$970$	0	0
$971$	−16.6842	−0.535420	−0.267710	−	0.963500i	$-0.586267\pi$
$971$	−16.6842	−0.535420	−0.267710	+	0.963500i	$0.586267\pi$
$972$	0	0
$973$	27.6046	0.884964
$974$	0	0
$975$	−42.8545	−1.37244
$976$	0	0
$977$	34.2205	1.09481	0.547405	−	0.836868i	$-0.315616\pi$
$977$	34.2205	1.09481	0.547405	+	0.836868i	$0.315616\pi$
$978$	0	0
$979$	−13.8246	−0.441837
$980$	0	0
$981$	8.58085	0.273966
$982$	0	0
$983$	−22.4852	−0.717165	−0.358583	−	0.933498i	$-0.616740\pi$
$983$	−22.4852	−0.717165	−0.358583	+	0.933498i	$0.616740\pi$
$984$	0	0
$985$	92.3638	2.94295
$986$	0	0
$987$	−57.7830	−1.83925
$988$	0	0
$989$	30.5197	0.970469
$990$	0	0
$991$	−2.27255	−0.0721898	−0.0360949	−	0.999348i	$-0.511492\pi$
$991$	−2.27255	−0.0721898	−0.0360949	+	0.999348i	$0.511492\pi$
$992$	0	0
$993$	85.6769	2.71887
$994$	0	0
$995$	−93.1015	−2.95152
$996$	0	0
$997$	−11.9501	−0.378463	−0.189232	−	0.981932i	$-0.560600\pi$
$997$	−11.9501	−0.378463	−0.189232	+	0.981932i	$0.560600\pi$
$998$	0	0
$999$	1.74735	0.0552837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.5	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.5	✓	49	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)

\(f(q)\)	\(=\)	\(q-2.55321 q^{3} -4.16598 q^{5} -2.08978 q^{7} +3.51888 q^{9} +O(q^{10})\)	\(q-2.55321 q^{3} -4.16598 q^{5} -2.08978 q^{7} +3.51888 q^{9} +1.07408 q^{11} +1.35848 q^{13} +10.6366 q^{15} +3.15848 q^{17} +1.85354 q^{19} +5.33566 q^{21} -3.48302 q^{23} +12.3554 q^{25} -1.32482 q^{27} -9.50723 q^{29} +8.19868 q^{31} -2.74235 q^{33} +8.70600 q^{35} -1.31894 q^{37} -3.46849 q^{39} -9.93589 q^{41} -8.76241 q^{43} -14.6596 q^{45} -10.8296 q^{47} -2.63280 q^{49} -8.06427 q^{51} -8.26073 q^{53} -4.47459 q^{55} -4.73248 q^{57} -3.05436 q^{59} -11.6854 q^{61} -7.35371 q^{63} -5.65941 q^{65} +9.10501 q^{67} +8.89289 q^{69} +6.64661 q^{71} +11.6562 q^{73} -31.5459 q^{75} -2.24459 q^{77} -3.49169 q^{79} -7.17411 q^{81} -12.0592 q^{83} -13.1582 q^{85} +24.2740 q^{87} -12.8712 q^{89} -2.83893 q^{91} -20.9330 q^{93} -7.72181 q^{95} +13.7435 q^{97} +3.77955 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 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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Database

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