LMFDB - Embedded newform 8008.2.a.r.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 $ x^9 - 4x^8 - 9x^7 + 36x^6 + 23x^5 - 89x^4 - 20x^3 + 51x^2 + 18x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.302339$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30234 q^{3} -2.00169 q^{5} -1.00000 q^{7} -1.30391 q^{9} +O(q^{10})$	$q+1.30234 q^{3} -2.00169 q^{5} -1.00000 q^{7} -1.30391 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.60688 q^{15} -5.26988 q^{17} -0.954789 q^{19} -1.30234 q^{21} -1.20310 q^{23} -0.993233 q^{25} -5.60515 q^{27} -2.14941 q^{29} +1.52626 q^{31} -1.30234 q^{33} +2.00169 q^{35} -5.62591 q^{37} -1.30234 q^{39} +9.27385 q^{41} +3.39860 q^{43} +2.61003 q^{45} -7.83643 q^{47} +1.00000 q^{49} -6.86317 q^{51} +10.8136 q^{53} +2.00169 q^{55} -1.24346 q^{57} +6.62315 q^{59} +13.6207 q^{61} +1.30391 q^{63} +2.00169 q^{65} -13.1341 q^{67} -1.56684 q^{69} +6.54959 q^{71} +2.70094 q^{73} -1.29353 q^{75} +1.00000 q^{77} +8.67942 q^{79} -3.38806 q^{81} +9.60234 q^{83} +10.5487 q^{85} -2.79926 q^{87} +0.995708 q^{89} +1.00000 q^{91} +1.98771 q^{93} +1.91119 q^{95} +1.55964 q^{97} +1.30391 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * 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
$3$	1.30234	0.751906	0.375953	−	0.926639i	$-0.377315\pi$
$3$	1.30234	0.751906	0.375953	+	0.926639i	$0.377315\pi$
$4$	0	0
$5$	−2.00169	−0.895183	−0.447592	−	0.894238i	$-0.647718\pi$
$5$	−2.00169	−0.895183	−0.447592	+	0.894238i	$0.647718\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.30391	−0.434638
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.60688	−0.673093
$16$	0	0
$17$	−5.26988	−1.27813	−0.639067	−	0.769151i	$-0.720679\pi$
$17$	−5.26988	−1.27813	−0.639067	+	0.769151i	$0.720679\pi$
$18$	0	0
$19$	−0.954789	−0.219044	−0.109522	−	0.993984i	$-0.534932\pi$
$19$	−0.954789	−0.219044	−0.109522	+	0.993984i	$0.534932\pi$
$20$	0	0
$21$	−1.30234	−0.284194
$22$	0	0
$23$	−1.20310	−0.250863	−0.125432	−	0.992102i	$-0.540032\pi$
$23$	−1.20310	−0.250863	−0.125432	+	0.992102i	$0.540032\pi$
$24$	0	0
$25$	−0.993233	−0.198647
$26$	0	0
$27$	−5.60515	−1.07871
$28$	0	0
$29$	−2.14941	−0.399135	−0.199568	−	0.979884i	$-0.563954\pi$
$29$	−2.14941	−0.399135	−0.199568	+	0.979884i	$0.563954\pi$
$30$	0	0
$31$	1.52626	0.274124	0.137062	−	0.990562i	$-0.456234\pi$
$31$	1.52626	0.274124	0.137062	+	0.990562i	$0.456234\pi$
$32$	0	0
$33$	−1.30234	−0.226708
$34$	0	0
$35$	2.00169	0.338348
$36$	0	0
$37$	−5.62591	−0.924893	−0.462447	−	0.886647i	$-0.653028\pi$
$37$	−5.62591	−0.924893	−0.462447	+	0.886647i	$0.653028\pi$
$38$	0	0
$39$	−1.30234	−0.208541
$40$	0	0
$41$	9.27385	1.44833	0.724166	−	0.689626i	$-0.242225\pi$
$41$	9.27385	1.44833	0.724166	+	0.689626i	$0.242225\pi$
$42$	0	0
$43$	3.39860	0.518282	0.259141	−	0.965839i	$-0.416561\pi$
$43$	3.39860	0.518282	0.259141	+	0.965839i	$0.416561\pi$
$44$	0	0
$45$	2.61003	0.389081
$46$	0	0
$47$	−7.83643	−1.14306	−0.571530	−	0.820581i	$-0.693650\pi$
$47$	−7.83643	−1.14306	−0.571530	+	0.820581i	$0.693650\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.86317	−0.961035
$52$	0	0
$53$	10.8136	1.48536	0.742680	−	0.669646i	$-0.233554\pi$
$53$	10.8136	1.48536	0.742680	+	0.669646i	$0.233554\pi$
$54$	0	0
$55$	2.00169	0.269908
$56$	0	0
$57$	−1.24346	−0.164700
$58$	0	0
$59$	6.62315	0.862261	0.431130	−	0.902290i	$-0.358115\pi$
$59$	6.62315	0.862261	0.431130	+	0.902290i	$0.358115\pi$
$60$	0	0
$61$	13.6207	1.74396	0.871978	−	0.489544i	$-0.162837\pi$
$61$	13.6207	1.74396	0.871978	+	0.489544i	$0.162837\pi$
$62$	0	0
$63$	1.30391	0.164278
$64$	0	0
$65$	2.00169	0.248279
$66$	0	0
$67$	−13.1341	−1.60459	−0.802293	−	0.596930i	$-0.796387\pi$
$67$	−13.1341	−1.60459	−0.802293	+	0.596930i	$0.796387\pi$
$68$	0	0
$69$	−1.56684	−0.188626
$70$	0	0
$71$	6.54959	0.777293	0.388647	−	0.921387i	$-0.372943\pi$
$71$	6.54959	0.777293	0.388647	+	0.921387i	$0.372943\pi$
$72$	0	0
$73$	2.70094	0.316121	0.158061	−	0.987429i	$-0.449476\pi$
$73$	2.70094	0.316121	0.158061	+	0.987429i	$0.449476\pi$
$74$	0	0
$75$	−1.29353	−0.149363
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	8.67942	0.976512	0.488256	−	0.872701i	$-0.337633\pi$
$79$	8.67942	0.976512	0.488256	+	0.872701i	$0.337633\pi$
$80$	0	0
$81$	−3.38806	−0.376452
$82$	0	0
$83$	9.60234	1.05399	0.526997	−	0.849867i	$-0.323318\pi$
$83$	9.60234	1.05399	0.526997	+	0.849867i	$0.323318\pi$
$84$	0	0
$85$	10.5487	1.14416
$86$	0	0
$87$	−2.79926	−0.300112
$88$	0	0
$89$	0.995708	0.105545	0.0527724	−	0.998607i	$-0.483194\pi$
$89$	0.995708	0.105545	0.0527724	+	0.998607i	$0.483194\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.98771	0.206115
$94$	0	0
$95$	1.91119	0.196084
$96$	0	0
$97$	1.55964	0.158357	0.0791786	−	0.996860i	$-0.474770\pi$
$97$	1.55964	0.158357	0.0791786	+	0.996860i	$0.474770\pi$
$98$	0	0
$99$	1.30391	0.131048
$100$	0	0
$101$	−7.58542	−0.754778	−0.377389	−	0.926055i	$-0.623178\pi$
$101$	−7.58542	−0.754778	−0.377389	+	0.926055i	$0.623178\pi$
$102$	0	0
$103$	6.90142	0.680017	0.340008	−	0.940422i	$-0.389570\pi$
$103$	6.90142	0.680017	0.340008	+	0.940422i	$0.389570\pi$
$104$	0	0
$105$	2.60688	0.254405
$106$	0	0
$107$	−0.401125	−0.0387782	−0.0193891	−	0.999812i	$-0.506172\pi$
$107$	−0.401125	−0.0387782	−0.0193891	+	0.999812i	$0.506172\pi$
$108$	0	0
$109$	−2.37856	−0.227824	−0.113912	−	0.993491i	$-0.536338\pi$
$109$	−2.37856	−0.227824	−0.113912	+	0.993491i	$0.536338\pi$
$110$	0	0
$111$	−7.32683	−0.695432
$112$	0	0
$113$	10.7449	1.01079	0.505396	−	0.862888i	$-0.331346\pi$
$113$	10.7449	1.01079	0.505396	+	0.862888i	$0.331346\pi$
$114$	0	0
$115$	2.40823	0.224569
$116$	0	0
$117$	1.30391	0.120547
$118$	0	0
$119$	5.26988	0.483089
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.0777	1.08901
$124$	0	0
$125$	11.9966	1.07301
$126$	0	0
$127$	−1.74693	−0.155015	−0.0775073	−	0.996992i	$-0.524696\pi$
$127$	−1.74693	−0.155015	−0.0775073	+	0.996992i	$0.524696\pi$
$128$	0	0
$129$	4.42613	0.389699
$130$	0	0
$131$	−10.8132	−0.944750	−0.472375	−	0.881398i	$-0.656603\pi$
$131$	−10.8132	−0.944750	−0.472375	+	0.881398i	$0.656603\pi$
$132$	0	0
$133$	0.954789	0.0827907
$134$	0	0
$135$	11.2198	0.965645
$136$	0	0
$137$	−0.782342	−0.0668400	−0.0334200	−	0.999441i	$-0.510640\pi$
$137$	−0.782342	−0.0668400	−0.0334200	+	0.999441i	$0.510640\pi$
$138$	0	0
$139$	0.437818	0.0371352	0.0185676	−	0.999828i	$-0.494089\pi$
$139$	0.437818	0.0371352	0.0185676	+	0.999828i	$0.494089\pi$
$140$	0	0
$141$	−10.2057	−0.859473
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	4.30245	0.357299
$146$	0	0
$147$	1.30234	0.107415
$148$	0	0
$149$	−2.41870	−0.198148	−0.0990740	−	0.995080i	$-0.531588\pi$
$149$	−2.41870	−0.198148	−0.0990740	+	0.995080i	$0.531588\pi$
$150$	0	0
$151$	−17.7870	−1.44749	−0.723743	−	0.690070i	$-0.757580\pi$
$151$	−17.7870	−1.44749	−0.723743	+	0.690070i	$0.757580\pi$
$152$	0	0
$153$	6.87147	0.555525
$154$	0	0
$155$	−3.05510	−0.245391
$156$	0	0
$157$	−20.4775	−1.63428	−0.817142	−	0.576436i	$-0.804443\pi$
$157$	−20.4775	−1.63428	−0.817142	+	0.576436i	$0.804443\pi$
$158$	0	0
$159$	14.0830	1.11685
$160$	0	0
$161$	1.20310	0.0948174
$162$	0	0
$163$	23.0106	1.80233	0.901165	−	0.433475i	$-0.142713\pi$
$163$	23.0106	1.80233	0.901165	+	0.433475i	$0.142713\pi$
$164$	0	0
$165$	2.60688	0.202945
$166$	0	0
$167$	6.92309	0.535725	0.267862	−	0.963457i	$-0.413683\pi$
$167$	6.92309	0.535725	0.267862	+	0.963457i	$0.413683\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.24496	0.0952047
$172$	0	0
$173$	−13.7192	−1.04305	−0.521526	−	0.853235i	$-0.674637\pi$
$173$	−13.7192	−1.04305	−0.521526	+	0.853235i	$0.674637\pi$
$174$	0	0
$175$	0.993233	0.0750813
$176$	0	0
$177$	8.62558	0.648339
$178$	0	0
$179$	10.9324	0.817126	0.408563	−	0.912730i	$-0.366030\pi$
$179$	10.9324	0.817126	0.408563	+	0.912730i	$0.366030\pi$
$180$	0	0
$181$	−6.38402	−0.474521	−0.237260	−	0.971446i	$-0.576249\pi$
$181$	−6.38402	−0.474521	−0.237260	+	0.971446i	$0.576249\pi$
$182$	0	0
$183$	17.7388	1.31129
$184$	0	0
$185$	11.2613	0.827949
$186$	0	0
$187$	5.26988	0.385372
$188$	0	0
$189$	5.60515	0.407715
$190$	0	0
$191$	−23.9811	−1.73521	−0.867605	−	0.497254i	$-0.834342\pi$
$191$	−23.9811	−1.73521	−0.867605	+	0.497254i	$0.834342\pi$
$192$	0	0
$193$	15.7661	1.13487	0.567433	−	0.823419i	$-0.307936\pi$
$193$	15.7661	1.13487	0.567433	+	0.823419i	$0.307936\pi$
$194$	0	0
$195$	2.60688	0.186683
$196$	0	0
$197$	14.7015	1.04744	0.523721	−	0.851890i	$-0.324544\pi$
$197$	14.7015	1.04744	0.523721	+	0.851890i	$0.324544\pi$
$198$	0	0
$199$	−10.7346	−0.760955	−0.380477	−	0.924790i	$-0.624240\pi$
$199$	−10.7346	−0.760955	−0.380477	+	0.924790i	$0.624240\pi$
$200$	0	0
$201$	−17.1051	−1.20650
$202$	0	0
$203$	2.14941	0.150859
$204$	0	0
$205$	−18.5634	−1.29652
$206$	0	0
$207$	1.56874	0.109035
$208$	0	0
$209$	0.954789	0.0660441
$210$	0	0
$211$	−9.33629	−0.642737	−0.321368	−	0.946954i	$-0.604143\pi$
$211$	−9.33629	−0.642737	−0.321368	+	0.946954i	$0.604143\pi$
$212$	0	0
$213$	8.52978	0.584451
$214$	0	0
$215$	−6.80296	−0.463958
$216$	0	0
$217$	−1.52626	−0.103609
$218$	0	0
$219$	3.51754	0.237693
$220$	0	0
$221$	5.26988	0.354490
$222$	0	0
$223$	−2.11061	−0.141337	−0.0706686	−	0.997500i	$-0.522513\pi$
$223$	−2.11061	−0.141337	−0.0706686	+	0.997500i	$0.522513\pi$
$224$	0	0
$225$	1.29509	0.0863394
$226$	0	0
$227$	24.1966	1.60598	0.802991	−	0.595991i	$-0.203241\pi$
$227$	24.1966	1.60598	0.802991	+	0.595991i	$0.203241\pi$
$228$	0	0
$229$	15.7026	1.03766	0.518828	−	0.854879i	$-0.326369\pi$
$229$	15.7026	1.03766	0.518828	+	0.854879i	$0.326369\pi$
$230$	0	0
$231$	1.30234	0.0856876
$232$	0	0
$233$	11.6059	0.760328	0.380164	−	0.924919i	$-0.375868\pi$
$233$	11.6059	0.760328	0.380164	+	0.924919i	$0.375868\pi$
$234$	0	0
$235$	15.6861	1.02325
$236$	0	0
$237$	11.3035	0.734244
$238$	0	0
$239$	−5.03751	−0.325849	−0.162925	−	0.986639i	$-0.552093\pi$
$239$	−5.03751	−0.325849	−0.162925	+	0.986639i	$0.552093\pi$
$240$	0	0
$241$	−29.2243	−1.88250	−0.941250	−	0.337711i	$-0.890347\pi$
$241$	−29.2243	−1.88250	−0.941250	+	0.337711i	$0.890347\pi$
$242$	0	0
$243$	12.4031	0.795656
$244$	0	0
$245$	−2.00169	−0.127883
$246$	0	0
$247$	0.954789	0.0607518
$248$	0	0
$249$	12.5055	0.792503
$250$	0	0
$251$	−12.2694	−0.774438	−0.387219	−	0.921988i	$-0.626564\pi$
$251$	−12.2694	−0.774438	−0.387219	+	0.921988i	$0.626564\pi$
$252$	0	0
$253$	1.20310	0.0756381
$254$	0	0
$255$	13.7379	0.860303
$256$	0	0
$257$	22.4534	1.40061	0.700304	−	0.713845i	$-0.253048\pi$
$257$	22.4534	1.40061	0.700304	+	0.713845i	$0.253048\pi$
$258$	0	0
$259$	5.62591	0.349577
$260$	0	0
$261$	2.80265	0.173479
$262$	0	0
$263$	−3.59826	−0.221879	−0.110939	−	0.993827i	$-0.535386\pi$
$263$	−3.59826	−0.221879	−0.110939	+	0.993827i	$0.535386\pi$
$264$	0	0
$265$	−21.6455	−1.32967
$266$	0	0
$267$	1.29675	0.0793597
$268$	0	0
$269$	−11.1729	−0.681222	−0.340611	−	0.940204i	$-0.610634\pi$
$269$	−11.1729	−0.681222	−0.340611	+	0.940204i	$0.610634\pi$
$270$	0	0
$271$	−12.4907	−0.758754	−0.379377	−	0.925242i	$-0.623862\pi$
$271$	−12.4907	−0.758754	−0.379377	+	0.925242i	$0.623862\pi$
$272$	0	0
$273$	1.30234	0.0788211
$274$	0	0
$275$	0.993233	0.0598942
$276$	0	0
$277$	23.0601	1.38555	0.692774	−	0.721155i	$-0.256388\pi$
$277$	23.0601	1.38555	0.692774	+	0.721155i	$0.256388\pi$
$278$	0	0
$279$	−1.99011	−0.119145
$280$	0	0
$281$	10.3534	0.617631	0.308816	−	0.951122i	$-0.400067\pi$
$281$	10.3534	0.617631	0.308816	+	0.951122i	$0.400067\pi$
$282$	0	0
$283$	−11.3106	−0.672342	−0.336171	−	0.941801i	$-0.609132\pi$
$283$	−11.3106	−0.672342	−0.336171	+	0.941801i	$0.609132\pi$
$284$	0	0
$285$	2.48902	0.147437
$286$	0	0
$287$	−9.27385	−0.547418
$288$	0	0
$289$	10.7716	0.633624
$290$	0	0
$291$	2.03117	0.119070
$292$	0	0
$293$	2.39569	0.139957	0.0699787	−	0.997548i	$-0.477707\pi$
$293$	2.39569	0.139957	0.0699787	+	0.997548i	$0.477707\pi$
$294$	0	0
$295$	−13.2575	−0.771882
$296$	0	0
$297$	5.60515	0.325244
$298$	0	0
$299$	1.20310	0.0695770
$300$	0	0
$301$	−3.39860	−0.195892
$302$	0	0
$303$	−9.87879	−0.567521
$304$	0	0
$305$	−27.2645	−1.56116
$306$	0	0
$307$	−15.9231	−0.908778	−0.454389	−	0.890803i	$-0.650142\pi$
$307$	−15.9231	−0.908778	−0.454389	+	0.890803i	$0.650142\pi$
$308$	0	0
$309$	8.98798	0.511308
$310$	0	0
$311$	26.3435	1.49380	0.746901	−	0.664935i	$-0.231541\pi$
$311$	26.3435	1.49380	0.746901	+	0.664935i	$0.231541\pi$
$312$	0	0
$313$	24.5904	1.38993	0.694966	−	0.719043i	$-0.255419\pi$
$313$	24.5904	1.38993	0.694966	+	0.719043i	$0.255419\pi$
$314$	0	0
$315$	−2.61003	−0.147059
$316$	0	0
$317$	9.36362	0.525913	0.262957	−	0.964808i	$-0.415302\pi$
$317$	9.36362	0.525913	0.262957	+	0.964808i	$0.415302\pi$
$318$	0	0
$319$	2.14941	0.120344
$320$	0	0
$321$	−0.522401	−0.0291576
$322$	0	0
$323$	5.03162	0.279967
$324$	0	0
$325$	0.993233	0.0550946
$326$	0	0
$327$	−3.09769	−0.171302
$328$	0	0
$329$	7.83643	0.432036
$330$	0	0
$331$	10.4701	0.575486	0.287743	−	0.957708i	$-0.407095\pi$
$331$	10.4701	0.575486	0.287743	+	0.957708i	$0.407095\pi$
$332$	0	0
$333$	7.33570	0.401994
$334$	0	0
$335$	26.2904	1.43640
$336$	0	0
$337$	32.4963	1.77018	0.885092	−	0.465416i	$-0.154095\pi$
$337$	32.4963	1.77018	0.885092	+	0.465416i	$0.154095\pi$
$338$	0	0
$339$	13.9934	0.760020
$340$	0	0
$341$	−1.52626	−0.0826515
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.13633	0.168854
$346$	0	0
$347$	14.3262	0.769071	0.384535	−	0.923110i	$-0.374362\pi$
$347$	14.3262	0.769071	0.384535	+	0.923110i	$0.374362\pi$
$348$	0	0
$349$	−18.4161	−0.985794	−0.492897	−	0.870088i	$-0.664062\pi$
$349$	−18.4161	−0.985794	−0.492897	+	0.870088i	$0.664062\pi$
$350$	0	0
$351$	5.60515	0.299181
$352$	0	0
$353$	10.3992	0.553491	0.276746	−	0.960943i	$-0.410744\pi$
$353$	10.3992	0.553491	0.276746	+	0.960943i	$0.410744\pi$
$354$	0	0
$355$	−13.1103	−0.695820
$356$	0	0
$357$	6.86317	0.363237
$358$	0	0
$359$	−4.04563	−0.213520	−0.106760	−	0.994285i	$-0.534048\pi$
$359$	−4.04563	−0.213520	−0.106760	+	0.994285i	$0.534048\pi$
$360$	0	0
$361$	−18.0884	−0.952020
$362$	0	0
$363$	1.30234	0.0683550
$364$	0	0
$365$	−5.40645	−0.282987
$366$	0	0
$367$	6.81681	0.355835	0.177917	−	0.984045i	$-0.443064\pi$
$367$	6.81681	0.355835	0.177917	+	0.984045i	$0.443064\pi$
$368$	0	0
$369$	−12.0923	−0.629500
$370$	0	0
$371$	−10.8136	−0.561414
$372$	0	0
$373$	−21.4594	−1.11112	−0.555562	−	0.831475i	$-0.687497\pi$
$373$	−21.4594	−1.11112	−0.555562	+	0.831475i	$0.687497\pi$
$374$	0	0
$375$	15.6236	0.806801
$376$	0	0
$377$	2.14941	0.110700
$378$	0	0
$379$	14.1422	0.726434	0.363217	−	0.931705i	$-0.381678\pi$
$379$	14.1422	0.726434	0.363217	+	0.931705i	$0.381678\pi$
$380$	0	0
$381$	−2.27509	−0.116556
$382$	0	0
$383$	15.9062	0.812767	0.406384	−	0.913703i	$-0.366790\pi$
$383$	15.9062	0.812767	0.406384	+	0.913703i	$0.366790\pi$
$384$	0	0
$385$	−2.00169	−0.102016
$386$	0	0
$387$	−4.43149	−0.225265
$388$	0	0
$389$	−29.2033	−1.48066	−0.740332	−	0.672241i	$-0.765332\pi$
$389$	−29.2033	−1.48066	−0.740332	+	0.672241i	$0.765332\pi$
$390$	0	0
$391$	6.34018	0.320637
$392$	0	0
$393$	−14.0824	−0.710363
$394$	0	0
$395$	−17.3735	−0.874157
$396$	0	0
$397$	−15.8205	−0.794008	−0.397004	−	0.917817i	$-0.629950\pi$
$397$	−15.8205	−0.794008	−0.397004	+	0.917817i	$0.629950\pi$
$398$	0	0
$399$	1.24346	0.0622508
$400$	0	0
$401$	12.9904	0.648709	0.324355	−	0.945936i	$-0.394853\pi$
$401$	12.9904	0.648709	0.324355	+	0.945936i	$0.394853\pi$
$402$	0	0
$403$	−1.52626	−0.0760284
$404$	0	0
$405$	6.78186	0.336993
$406$	0	0
$407$	5.62591	0.278866
$408$	0	0
$409$	−19.4368	−0.961086	−0.480543	−	0.876971i	$-0.659560\pi$
$409$	−19.4368	−0.961086	−0.480543	+	0.876971i	$0.659560\pi$
$410$	0	0
$411$	−1.01887	−0.0502574
$412$	0	0
$413$	−6.62315	−0.325904
$414$	0	0
$415$	−19.2209	−0.943518
$416$	0	0
$417$	0.570187	0.0279222
$418$	0	0
$419$	31.7609	1.55162	0.775811	−	0.630965i	$-0.217341\pi$
$419$	31.7609	1.55162	0.775811	+	0.630965i	$0.217341\pi$
$420$	0	0
$421$	31.3025	1.52559	0.762794	−	0.646641i	$-0.223827\pi$
$421$	31.3025	1.52559	0.762794	+	0.646641i	$0.223827\pi$
$422$	0	0
$423$	10.2180	0.496818
$424$	0	0
$425$	5.23422	0.253897
$426$	0	0
$427$	−13.6207	−0.659154
$428$	0	0
$429$	1.30234	0.0628775
$430$	0	0
$431$	18.9838	0.914418	0.457209	−	0.889359i	$-0.348849\pi$
$431$	18.9838	0.914418	0.457209	+	0.889359i	$0.348849\pi$
$432$	0	0
$433$	−22.9953	−1.10508	−0.552541	−	0.833486i	$-0.686342\pi$
$433$	−22.9953	−1.10508	−0.552541	+	0.833486i	$0.686342\pi$
$434$	0	0
$435$	5.60325	0.268655
$436$	0	0
$437$	1.14870	0.0549500
$438$	0	0
$439$	27.0664	1.29181	0.645903	−	0.763419i	$-0.276481\pi$
$439$	27.0664	1.29181	0.645903	+	0.763419i	$0.276481\pi$
$440$	0	0
$441$	−1.30391	−0.0620912
$442$	0	0
$443$	−6.48235	−0.307986	−0.153993	−	0.988072i	$-0.549213\pi$
$443$	−6.48235	−0.307986	−0.153993	+	0.988072i	$0.549213\pi$
$444$	0	0
$445$	−1.99310	−0.0944820
$446$	0	0
$447$	−3.14997	−0.148989
$448$	0	0
$449$	21.0017	0.991132	0.495566	−	0.868570i	$-0.334961\pi$
$449$	21.0017	0.991132	0.495566	+	0.868570i	$0.334961\pi$
$450$	0	0
$451$	−9.27385	−0.436689
$452$	0	0
$453$	−23.1647	−1.08837
$454$	0	0
$455$	−2.00169	−0.0938407
$456$	0	0
$457$	−10.2648	−0.480168	−0.240084	−	0.970752i	$-0.577175\pi$
$457$	−10.2648	−0.480168	−0.240084	+	0.970752i	$0.577175\pi$
$458$	0	0
$459$	29.5385	1.37874
$460$	0	0
$461$	16.2513	0.756901	0.378450	−	0.925622i	$-0.376457\pi$
$461$	16.2513	0.756901	0.378450	+	0.925622i	$0.376457\pi$
$462$	0	0
$463$	28.0049	1.30150	0.650749	−	0.759293i	$-0.274455\pi$
$463$	28.0049	1.30150	0.650749	+	0.759293i	$0.274455\pi$
$464$	0	0
$465$	−3.97877	−0.184511
$466$	0	0
$467$	4.13967	0.191561	0.0957804	−	0.995402i	$-0.469465\pi$
$467$	4.13967	0.191561	0.0957804	+	0.995402i	$0.469465\pi$
$468$	0	0
$469$	13.1341	0.606477
$470$	0	0
$471$	−26.6687	−1.22883
$472$	0	0
$473$	−3.39860	−0.156268
$474$	0	0
$475$	0.948328	0.0435123
$476$	0	0
$477$	−14.1000	−0.645594
$478$	0	0
$479$	26.1182	1.19337	0.596686	−	0.802475i	$-0.296484\pi$
$479$	26.1182	1.19337	0.596686	+	0.802475i	$0.296484\pi$
$480$	0	0
$481$	5.62591	0.256519
$482$	0	0
$483$	1.56684	0.0712937
$484$	0	0
$485$	−3.12191	−0.141759
$486$	0	0
$487$	9.18690	0.416298	0.208149	−	0.978097i	$-0.433256\pi$
$487$	9.18690	0.416298	0.208149	+	0.978097i	$0.433256\pi$
$488$	0	0
$489$	29.9676	1.35518
$490$	0	0
$491$	0.992451	0.0447887	0.0223943	−	0.999749i	$-0.492871\pi$
$491$	0.992451	0.0447887	0.0223943	+	0.999749i	$0.492871\pi$
$492$	0	0
$493$	11.3271	0.510148
$494$	0	0
$495$	−2.61003	−0.117312
$496$	0	0
$497$	−6.54959	−0.293789
$498$	0	0
$499$	−24.5957	−1.10105	−0.550527	−	0.834817i	$-0.685573\pi$
$499$	−24.5957	−1.10105	−0.550527	+	0.834817i	$0.685573\pi$
$500$	0	0
$501$	9.01620	0.402814
$502$	0	0
$503$	−23.0573	−1.02808	−0.514038	−	0.857767i	$-0.671851\pi$
$503$	−23.0573	−1.02808	−0.514038	+	0.857767i	$0.671851\pi$
$504$	0	0
$505$	15.1837	0.675664
$506$	0	0
$507$	1.30234	0.0578389
$508$	0	0
$509$	−1.43379	−0.0635516	−0.0317758	−	0.999495i	$-0.510116\pi$
$509$	−1.43379	−0.0635516	−0.0317758	+	0.999495i	$0.510116\pi$
$510$	0	0
$511$	−2.70094	−0.119483
$512$	0	0
$513$	5.35174	0.236285
$514$	0	0
$515$	−13.8145	−0.608740
$516$	0	0
$517$	7.83643	0.344646
$518$	0	0
$519$	−17.8671	−0.784277
$520$	0	0
$521$	−13.7715	−0.603340	−0.301670	−	0.953412i	$-0.597544\pi$
$521$	−13.7715	−0.603340	−0.301670	+	0.953412i	$0.597544\pi$
$522$	0	0
$523$	38.6577	1.69038	0.845191	−	0.534464i	$-0.179487\pi$
$523$	38.6577	1.69038	0.845191	+	0.534464i	$0.179487\pi$
$524$	0	0
$525$	1.29353	0.0564541
$526$	0	0
$527$	−8.04320	−0.350367
$528$	0	0
$529$	−21.5526	−0.937068
$530$	0	0
$531$	−8.63602	−0.374771
$532$	0	0
$533$	−9.27385	−0.401695
$534$	0	0
$535$	0.802929	0.0347136
$536$	0	0
$537$	14.2377	0.614402
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−16.6295	−0.714957	−0.357478	−	0.933921i	$-0.616363\pi$
$541$	−16.6295	−0.714957	−0.357478	+	0.933921i	$0.616363\pi$
$542$	0	0
$543$	−8.31416	−0.356795
$544$	0	0
$545$	4.76113	0.203945
$546$	0	0
$547$	3.05703	0.130709	0.0653545	−	0.997862i	$-0.479182\pi$
$547$	3.05703	0.130709	0.0653545	+	0.997862i	$0.479182\pi$
$548$	0	0
$549$	−17.7603	−0.757990
$550$	0	0
$551$	2.05223	0.0874280
$552$	0	0
$553$	−8.67942	−0.369087
$554$	0	0
$555$	14.6661	0.622540
$556$	0	0
$557$	−19.2744	−0.816681	−0.408341	−	0.912830i	$-0.633892\pi$
$557$	−19.2744	−0.816681	−0.408341	+	0.912830i	$0.633892\pi$
$558$	0	0
$559$	−3.39860	−0.143746
$560$	0	0
$561$	6.86317	0.289763
$562$	0	0
$563$	7.35177	0.309840	0.154920	−	0.987927i	$-0.450488\pi$
$563$	7.35177	0.309840	0.154920	+	0.987927i	$0.450488\pi$
$564$	0	0
$565$	−21.5079	−0.904844
$566$	0	0
$567$	3.38806	0.142285
$568$	0	0
$569$	−7.80152	−0.327057	−0.163528	−	0.986539i	$-0.552288\pi$
$569$	−7.80152	−0.327057	−0.163528	+	0.986539i	$0.552288\pi$
$570$	0	0
$571$	9.15113	0.382963	0.191482	−	0.981496i	$-0.438671\pi$
$571$	9.15113	0.382963	0.191482	+	0.981496i	$0.438671\pi$
$572$	0	0
$573$	−31.2315	−1.30471
$574$	0	0
$575$	1.19496	0.0498331
$576$	0	0
$577$	−42.4873	−1.76877	−0.884386	−	0.466757i	$-0.845422\pi$
$577$	−42.4873	−1.76877	−0.884386	+	0.466757i	$0.845422\pi$
$578$	0	0
$579$	20.5328	0.853313
$580$	0	0
$581$	−9.60234	−0.398372
$582$	0	0
$583$	−10.8136	−0.447853
$584$	0	0
$585$	−2.61003	−0.107912
$586$	0	0
$587$	34.0255	1.40438	0.702192	−	0.711988i	$-0.252205\pi$
$587$	34.0255	1.40438	0.702192	+	0.711988i	$0.252205\pi$
$588$	0	0
$589$	−1.45725	−0.0600451
$590$	0	0
$591$	19.1464	0.787577
$592$	0	0
$593$	−10.1978	−0.418775	−0.209388	−	0.977833i	$-0.567147\pi$
$593$	−10.1978	−0.418775	−0.209388	+	0.977833i	$0.567147\pi$
$594$	0	0
$595$	−10.5487	−0.432453
$596$	0	0
$597$	−13.9801	−0.572166
$598$	0	0
$599$	−46.2930	−1.89148	−0.945741	−	0.324921i	$-0.894662\pi$
$599$	−46.2930	−1.89148	−0.945741	+	0.324921i	$0.894662\pi$
$600$	0	0
$601$	26.2013	1.06877	0.534387	−	0.845240i	$-0.320543\pi$
$601$	26.2013	1.06877	0.534387	+	0.845240i	$0.320543\pi$
$602$	0	0
$603$	17.1258	0.697415
$604$	0	0
$605$	−2.00169	−0.0813803
$606$	0	0
$607$	−20.1402	−0.817466	−0.408733	−	0.912654i	$-0.634029\pi$
$607$	−20.1402	−0.817466	−0.408733	+	0.912654i	$0.634029\pi$
$608$	0	0
$609$	2.79926	0.113432
$610$	0	0
$611$	7.83643	0.317028
$612$	0	0
$613$	25.5956	1.03380	0.516899	−	0.856047i	$-0.327086\pi$
$613$	25.5956	1.03380	0.516899	+	0.856047i	$0.327086\pi$
$614$	0	0
$615$	−24.1758	−0.974863
$616$	0	0
$617$	31.6950	1.27599	0.637995	−	0.770040i	$-0.279764\pi$
$617$	31.6950	1.27599	0.637995	+	0.770040i	$0.279764\pi$
$618$	0	0
$619$	−3.55058	−0.142710	−0.0713550	−	0.997451i	$-0.522732\pi$
$619$	−3.55058	−0.142710	−0.0713550	+	0.997451i	$0.522732\pi$
$620$	0	0
$621$	6.74355	0.270609
$622$	0	0
$623$	−0.995708	−0.0398922
$624$	0	0
$625$	−19.0473	−0.761893
$626$	0	0
$627$	1.24346	0.0496589
$628$	0	0
$629$	29.6478	1.18214
$630$	0	0
$631$	29.4885	1.17392	0.586960	−	0.809616i	$-0.300325\pi$
$631$	29.4885	1.17392	0.586960	+	0.809616i	$0.300325\pi$
$632$	0	0
$633$	−12.1590	−0.483277
$634$	0	0
$635$	3.49680	0.138766
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−8.54010	−0.337841
$640$	0	0
$641$	27.5333	1.08750	0.543751	−	0.839247i	$-0.317004\pi$
$641$	27.5333	1.08750	0.543751	+	0.839247i	$0.317004\pi$
$642$	0	0
$643$	35.4318	1.39729	0.698647	−	0.715467i	$-0.253786\pi$
$643$	35.4318	1.39729	0.698647	+	0.715467i	$0.253786\pi$
$644$	0	0
$645$	−8.85975	−0.348852
$646$	0	0
$647$	−4.65328	−0.182939	−0.0914697	−	0.995808i	$-0.529156\pi$
$647$	−4.65328	−0.182939	−0.0914697	+	0.995808i	$0.529156\pi$
$648$	0	0
$649$	−6.62315	−0.259981
$650$	0	0
$651$	−1.98771	−0.0779043
$652$	0	0
$653$	50.6095	1.98050	0.990251	−	0.139297i	$-0.0444842\pi$
$653$	50.6095	1.98050	0.990251	+	0.139297i	$0.0444842\pi$
$654$	0	0
$655$	21.6446	0.845724
$656$	0	0
$657$	−3.52180	−0.137398
$658$	0	0
$659$	−42.7094	−1.66372	−0.831860	−	0.554985i	$-0.812724\pi$
$659$	−42.7094	−1.66372	−0.831860	+	0.554985i	$0.812724\pi$
$660$	0	0
$661$	−12.4240	−0.483237	−0.241618	−	0.970371i	$-0.577678\pi$
$661$	−12.4240	−0.483237	−0.241618	+	0.970371i	$0.577678\pi$
$662$	0	0
$663$	6.86317	0.266543
$664$	0	0
$665$	−1.91119	−0.0741129
$666$	0	0
$667$	2.58595	0.100128
$668$	0	0
$669$	−2.74873	−0.106272
$670$	0	0
$671$	−13.6207	−0.525823
$672$	0	0
$673$	−16.1983	−0.624398	−0.312199	−	0.950017i	$-0.601066\pi$
$673$	−16.1983	−0.624398	−0.312199	+	0.950017i	$0.601066\pi$
$674$	0	0
$675$	5.56722	0.214282
$676$	0	0
$677$	−36.5801	−1.40589	−0.702943	−	0.711246i	$-0.748131\pi$
$677$	−36.5801	−1.40589	−0.702943	+	0.711246i	$0.748131\pi$
$678$	0	0
$679$	−1.55964	−0.0598534
$680$	0	0
$681$	31.5121	1.20755
$682$	0	0
$683$	0.421928	0.0161446	0.00807231	−	0.999967i	$-0.497430\pi$
$683$	0.421928	0.0161446	0.00807231	+	0.999967i	$0.497430\pi$
$684$	0	0
$685$	1.56601	0.0598341
$686$	0	0
$687$	20.4501	0.780219
$688$	0	0
$689$	−10.8136	−0.411965
$690$	0	0
$691$	19.3717	0.736936	0.368468	−	0.929641i	$-0.379882\pi$
$691$	19.3717	0.736936	0.368468	+	0.929641i	$0.379882\pi$
$692$	0	0
$693$	−1.30391	−0.0495316
$694$	0	0
$695$	−0.876376	−0.0332428
$696$	0	0
$697$	−48.8721	−1.85116
$698$	0	0
$699$	15.1148	0.571694
$700$	0	0
$701$	8.64988	0.326701	0.163351	−	0.986568i	$-0.447770\pi$
$701$	8.64988	0.326701	0.163351	+	0.986568i	$0.447770\pi$
$702$	0	0
$703$	5.37155	0.202592
$704$	0	0
$705$	20.4286	0.769386
$706$	0	0
$707$	7.58542	0.285279
$708$	0	0
$709$	−33.1487	−1.24493	−0.622463	−	0.782649i	$-0.713868\pi$
$709$	−33.1487	−1.24493	−0.622463	+	0.782649i	$0.713868\pi$
$710$	0	0
$711$	−11.3172	−0.424429
$712$	0	0
$713$	−1.83624	−0.0687677
$714$	0	0
$715$	−2.00169	−0.0748590
$716$	0	0
$717$	−6.56054	−0.245008
$718$	0	0
$719$	2.46720	0.0920109	0.0460054	−	0.998941i	$-0.485351\pi$
$719$	2.46720	0.0920109	0.0460054	+	0.998941i	$0.485351\pi$
$720$	0	0
$721$	−6.90142	−0.257022
$722$	0	0
$723$	−38.0599	−1.41546
$724$	0	0
$725$	2.13486	0.0792869
$726$	0	0
$727$	−15.5544	−0.576880	−0.288440	−	0.957498i	$-0.593137\pi$
$727$	−15.5544	−0.576880	−0.288440	+	0.957498i	$0.593137\pi$
$728$	0	0
$729$	26.3172	0.974710
$730$	0	0
$731$	−17.9102	−0.662434
$732$	0	0
$733$	−25.8539	−0.954936	−0.477468	−	0.878649i	$-0.658445\pi$
$733$	−25.8539	−0.954936	−0.477468	+	0.878649i	$0.658445\pi$
$734$	0	0
$735$	−2.60688	−0.0961562
$736$	0	0
$737$	13.1341	0.483801
$738$	0	0
$739$	−2.12055	−0.0780058	−0.0390029	−	0.999239i	$-0.512418\pi$
$739$	−2.12055	−0.0780058	−0.0390029	+	0.999239i	$0.512418\pi$
$740$	0	0
$741$	1.24346	0.0456796
$742$	0	0
$743$	0.174751	0.00641098	0.00320549	−	0.999995i	$-0.498980\pi$
$743$	0.174751	0.00641098	0.00320549	+	0.999995i	$0.498980\pi$
$744$	0	0
$745$	4.84150	0.177379
$746$	0	0
$747$	−12.5206	−0.458106
$748$	0	0
$749$	0.401125	0.0146568
$750$	0	0
$751$	−35.0419	−1.27870	−0.639349	−	0.768917i	$-0.720796\pi$
$751$	−35.0419	−1.27870	−0.639349	+	0.768917i	$0.720796\pi$
$752$	0	0
$753$	−15.9789	−0.582305
$754$	0	0
$755$	35.6041	1.29577
$756$	0	0
$757$	25.9566	0.943408	0.471704	−	0.881757i	$-0.343639\pi$
$757$	25.9566	0.943408	0.471704	+	0.881757i	$0.343639\pi$
$758$	0	0
$759$	1.56684	0.0568727
$760$	0	0
$761$	−43.8848	−1.59082	−0.795412	−	0.606069i	$-0.792745\pi$
$761$	−43.8848	−1.59082	−0.795412	+	0.606069i	$0.792745\pi$
$762$	0	0
$763$	2.37856	0.0861095
$764$	0	0
$765$	−13.7546	−0.497297
$766$	0	0
$767$	−6.62315	−0.239148
$768$	0	0
$769$	1.98153	0.0714556	0.0357278	−	0.999362i	$-0.488625\pi$
$769$	1.98153	0.0714556	0.0357278	+	0.999362i	$0.488625\pi$
$770$	0	0
$771$	29.2420	1.05312
$772$	0	0
$773$	12.1098	0.435561	0.217780	−	0.975998i	$-0.430118\pi$
$773$	12.1098	0.435561	0.217780	+	0.975998i	$0.430118\pi$
$774$	0	0
$775$	−1.51593	−0.0544538
$776$	0	0
$777$	7.32683	0.262849
$778$	0	0
$779$	−8.85457	−0.317248
$780$	0	0
$781$	−6.54959	−0.234363
$782$	0	0
$783$	12.0478	0.430552
$784$	0	0
$785$	40.9897	1.46298
$786$	0	0
$787$	4.01971	0.143287	0.0716436	−	0.997430i	$-0.477176\pi$
$787$	4.01971	0.143287	0.0716436	+	0.997430i	$0.477176\pi$
$788$	0	0
$789$	−4.68616	−0.166832
$790$	0	0
$791$	−10.7449	−0.382043
$792$	0	0
$793$	−13.6207	−0.483687
$794$	0	0
$795$	−28.1897	−0.999786
$796$	0	0
$797$	−50.2199	−1.77888	−0.889439	−	0.457053i	$-0.848905\pi$
$797$	−50.2199	−1.77888	−0.889439	+	0.457053i	$0.848905\pi$
$798$	0	0
$799$	41.2970	1.46098
$800$	0	0
$801$	−1.29832	−0.0458738
$802$	0	0
$803$	−2.70094	−0.0953142
$804$	0	0
$805$	−2.40823	−0.0848790
$806$	0	0
$807$	−14.5509	−0.512214
$808$	0	0
$809$	−0.603691	−0.0212246	−0.0106123	−	0.999944i	$-0.503378\pi$
$809$	−0.603691	−0.0212246	−0.0106123	+	0.999944i	$0.503378\pi$
$810$	0	0
$811$	49.8307	1.74979	0.874896	−	0.484312i	$-0.160930\pi$
$811$	49.8307	1.74979	0.874896	+	0.484312i	$0.160930\pi$
$812$	0	0
$813$	−16.2671	−0.570511
$814$	0	0
$815$	−46.0602	−1.61342
$816$	0	0
$817$	−3.24495	−0.113526
$818$	0	0
$819$	−1.30391	−0.0455625
$820$	0	0
$821$	−48.9510	−1.70840	−0.854201	−	0.519943i	$-0.825953\pi$
$821$	−48.9510	−1.70840	−0.854201	+	0.519943i	$0.825953\pi$
$822$	0	0
$823$	−39.5896	−1.38001	−0.690004	−	0.723806i	$-0.742391\pi$
$823$	−39.5896	−1.38001	−0.690004	+	0.723806i	$0.742391\pi$
$824$	0	0
$825$	1.29353	0.0450348
$826$	0	0
$827$	23.9986	0.834514	0.417257	−	0.908788i	$-0.362991\pi$
$827$	23.9986	0.834514	0.417257	+	0.908788i	$0.362991\pi$
$828$	0	0
$829$	−29.4424	−1.02258	−0.511288	−	0.859409i	$-0.670831\pi$
$829$	−29.4424	−1.02258	−0.511288	+	0.859409i	$0.670831\pi$
$830$	0	0
$831$	30.0321	1.04180
$832$	0	0
$833$	−5.26988	−0.182590
$834$	0	0
$835$	−13.8579	−0.479572
$836$	0	0
$837$	−8.55491	−0.295701
$838$	0	0
$839$	−5.11424	−0.176563	−0.0882815	−	0.996096i	$-0.528138\pi$
$839$	−5.11424	−0.176563	−0.0882815	+	0.996096i	$0.528138\pi$
$840$	0	0
$841$	−24.3800	−0.840691
$842$	0	0
$843$	13.4836	0.464400
$844$	0	0
$845$	−2.00169	−0.0688603
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−14.7302	−0.505538
$850$	0	0
$851$	6.76852	0.232022
$852$	0	0
$853$	−43.1897	−1.47879	−0.739394	−	0.673273i	$-0.764888\pi$
$853$	−43.1897	−1.47879	−0.739394	+	0.673273i	$0.764888\pi$
$854$	0	0
$855$	−2.49203	−0.0852257
$856$	0	0
$857$	−22.3892	−0.764800	−0.382400	−	0.923997i	$-0.624902\pi$
$857$	−22.3892	−0.764800	−0.382400	+	0.923997i	$0.624902\pi$
$858$	0	0
$859$	26.4949	0.903995	0.451997	−	0.892019i	$-0.350712\pi$
$859$	26.4949	0.903995	0.451997	+	0.892019i	$0.350712\pi$
$860$	0	0
$861$	−12.0777	−0.411607
$862$	0	0
$863$	6.52482	0.222107	0.111054	−	0.993814i	$-0.464577\pi$
$863$	6.52482	0.222107	0.111054	+	0.993814i	$0.464577\pi$
$864$	0	0
$865$	27.4616	0.933723
$866$	0	0
$867$	14.0283	0.476426
$868$	0	0
$869$	−8.67942	−0.294429
$870$	0	0
$871$	13.1341	0.445032
$872$	0	0
$873$	−2.03363	−0.0688280
$874$	0	0
$875$	−11.9966	−0.405559
$876$	0	0
$877$	19.0844	0.644435	0.322218	−	0.946666i	$-0.395572\pi$
$877$	19.0844	0.644435	0.322218	+	0.946666i	$0.395572\pi$
$878$	0	0
$879$	3.11999	0.105235
$880$	0	0
$881$	−26.9372	−0.907536	−0.453768	−	0.891120i	$-0.649921\pi$
$881$	−26.9372	−0.907536	−0.453768	+	0.891120i	$0.649921\pi$
$882$	0	0
$883$	34.2012	1.15096	0.575481	−	0.817815i	$-0.304815\pi$
$883$	34.2012	1.15096	0.575481	+	0.817815i	$0.304815\pi$
$884$	0	0
$885$	−17.2658	−0.580382
$886$	0	0
$887$	−13.4474	−0.451518	−0.225759	−	0.974183i	$-0.572486\pi$
$887$	−13.4474	−0.451518	−0.225759	+	0.974183i	$0.572486\pi$
$888$	0	0
$889$	1.74693	0.0585900
$890$	0	0
$891$	3.38806	0.113504
$892$	0	0
$893$	7.48213	0.250380
$894$	0	0
$895$	−21.8833	−0.731478
$896$	0	0
$897$	1.56684	0.0523153
$898$	0	0
$899$	−3.28055	−0.109413
$900$	0	0
$901$	−56.9863	−1.89849
$902$	0	0
$903$	−4.42613	−0.147292
$904$	0	0
$905$	12.7788	0.424783
$906$	0	0
$907$	45.0843	1.49700	0.748500	−	0.663135i	$-0.230774\pi$
$907$	45.0843	1.49700	0.748500	+	0.663135i	$0.230774\pi$
$908$	0	0
$909$	9.89074	0.328055
$910$	0	0
$911$	−6.93630	−0.229810	−0.114905	−	0.993376i	$-0.536656\pi$
$911$	−6.93630	−0.229810	−0.114905	+	0.993376i	$0.536656\pi$
$912$	0	0
$913$	−9.60234	−0.317791
$914$	0	0
$915$	−35.5076	−1.17385
$916$	0	0
$917$	10.8132	0.357082
$918$	0	0
$919$	−2.77798	−0.0916370	−0.0458185	−	0.998950i	$-0.514590\pi$
$919$	−2.77798	−0.0916370	−0.0458185	+	0.998950i	$0.514590\pi$
$920$	0	0
$921$	−20.7372	−0.683316
$922$	0	0
$923$	−6.54959	−0.215582
$924$	0	0
$925$	5.58783	0.183727
$926$	0	0
$927$	−8.99885	−0.295561
$928$	0	0
$929$	40.0741	1.31479	0.657394	−	0.753547i	$-0.271659\pi$
$929$	40.0741	1.31479	0.657394	+	0.753547i	$0.271659\pi$
$930$	0	0
$931$	−0.954789	−0.0312919
$932$	0	0
$933$	34.3081	1.12320
$934$	0	0
$935$	−10.5487	−0.344978
$936$	0	0
$937$	−1.49076	−0.0487010	−0.0243505	−	0.999703i	$-0.507752\pi$
$937$	−1.49076	−0.0487010	−0.0243505	+	0.999703i	$0.507752\pi$
$938$	0	0
$939$	32.0250	1.04510
$940$	0	0
$941$	41.8411	1.36398	0.681991	−	0.731360i	$-0.261114\pi$
$941$	41.8411	1.36398	0.681991	+	0.731360i	$0.261114\pi$
$942$	0	0
$943$	−11.1574	−0.363333
$944$	0	0
$945$	−11.2198	−0.364980
$946$	0	0
$947$	9.77740	0.317723	0.158861	−	0.987301i	$-0.449218\pi$
$947$	9.77740	0.317723	0.158861	+	0.987301i	$0.449218\pi$
$948$	0	0
$949$	−2.70094	−0.0876763
$950$	0	0
$951$	12.1946	0.395437
$952$	0	0
$953$	41.9627	1.35930	0.679652	−	0.733535i	$-0.262131\pi$
$953$	41.9627	1.35930	0.679652	+	0.733535i	$0.262131\pi$
$954$	0	0
$955$	48.0027	1.55333
$956$	0	0
$957$	2.79926	0.0904872
$958$	0	0
$959$	0.782342	0.0252631
$960$	0	0
$961$	−28.6705	−0.924856
$962$	0	0
$963$	0.523033	0.0168545
$964$	0	0
$965$	−31.5588	−1.01591
$966$	0	0
$967$	25.1013	0.807204	0.403602	−	0.914935i	$-0.367758\pi$
$967$	25.1013	0.807204	0.403602	+	0.914935i	$0.367758\pi$
$968$	0	0
$969$	6.55287	0.210509
$970$	0	0
$971$	17.9037	0.574557	0.287278	−	0.957847i	$-0.407250\pi$
$971$	17.9037	0.574557	0.287278	+	0.957847i	$0.407250\pi$
$972$	0	0
$973$	−0.437818	−0.0140358
$974$	0	0
$975$	1.29353	0.0414260
$976$	0	0
$977$	33.3859	1.06811	0.534054	−	0.845450i	$-0.320668\pi$
$977$	33.3859	1.06811	0.534054	+	0.845450i	$0.320668\pi$
$978$	0	0
$979$	−0.995708	−0.0318230
$980$	0	0
$981$	3.10143	0.0990211
$982$	0	0
$983$	−0.236087	−0.00752999	−0.00376500	−	0.999993i	$-0.501198\pi$
$983$	−0.236087	−0.00752999	−0.00376500	+	0.999993i	$0.501198\pi$
$984$	0	0
$985$	−29.4279	−0.937652
$986$	0	0
$987$	10.2057	0.324850
$988$	0	0
$989$	−4.08885	−0.130018
$990$	0	0
$991$	−2.54608	−0.0808789	−0.0404395	−	0.999182i	$-0.512876\pi$
$991$	−2.54608	−0.0808789	−0.0404395	+	0.999182i	$0.512876\pi$
$992$	0	0
$993$	13.6356	0.432711
$994$	0	0
$995$	21.4873	0.681194
$996$	0	0
$997$	−15.5400	−0.492156	−0.246078	−	0.969250i	$-0.579142\pi$
$997$	−15.5400	−0.492156	−0.246078	+	0.969250i	$0.579142\pi$
$998$	0	0
$999$	31.5341	0.997694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.r.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.r.1.6	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30234 q^{3} -2.00169 q^{5} -1.00000 q^{7} -1.30391 q^{9} +O(q^{10})\)	\(q+1.30234 q^{3} -2.00169 q^{5} -1.00000 q^{7} -1.30391 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.60688 q^{15} -5.26988 q^{17} -0.954789 q^{19} -1.30234 q^{21} -1.20310 q^{23} -0.993233 q^{25} -5.60515 q^{27} -2.14941 q^{29} +1.52626 q^{31} -1.30234 q^{33} +2.00169 q^{35} -5.62591 q^{37} -1.30234 q^{39} +9.27385 q^{41} +3.39860 q^{43} +2.61003 q^{45} -7.83643 q^{47} +1.00000 q^{49} -6.86317 q^{51} +10.8136 q^{53} +2.00169 q^{55} -1.24346 q^{57} +6.62315 q^{59} +13.6207 q^{61} +1.30391 q^{63} +2.00169 q^{65} -13.1341 q^{67} -1.56684 q^{69} +6.54959 q^{71} +2.70094 q^{73} -1.29353 q^{75} +1.00000 q^{77} +8.67942 q^{79} -3.38806 q^{81} +9.60234 q^{83} +10.5487 q^{85} -2.79926 q^{87} +0.995708 q^{89} +1.00000 q^{91} +1.98771 q^{93} +1.91119 q^{95} +1.55964 q^{97} +1.30391 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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