LMFDB - Embedded newform 8008.2.a.z.1.4

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.04397$ 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)

$f(q)$	$=$	$q-2.04397 q^{3} +4.25466 q^{5} -1.00000 q^{7} +1.17780 q^{9} +O(q^{10})$	$q-2.04397 q^{3} +4.25466 q^{5} -1.00000 q^{7} +1.17780 q^{9} -1.00000 q^{11} +1.00000 q^{13} -8.69639 q^{15} -2.35304 q^{17} +5.32087 q^{19} +2.04397 q^{21} +3.35219 q^{23} +13.1022 q^{25} +3.72452 q^{27} -0.720205 q^{29} -9.19263 q^{31} +2.04397 q^{33} -4.25466 q^{35} +7.52534 q^{37} -2.04397 q^{39} +1.34603 q^{41} -5.57303 q^{43} +5.01114 q^{45} -5.81408 q^{47} +1.00000 q^{49} +4.80954 q^{51} +3.10736 q^{53} -4.25466 q^{55} -10.8757 q^{57} +2.11450 q^{59} +9.70962 q^{61} -1.17780 q^{63} +4.25466 q^{65} -5.48070 q^{67} -6.85177 q^{69} -7.12854 q^{71} -0.258610 q^{73} -26.7804 q^{75} +1.00000 q^{77} +6.57983 q^{79} -11.1462 q^{81} +13.7748 q^{83} -10.0114 q^{85} +1.47207 q^{87} -3.05729 q^{89} -1.00000 q^{91} +18.7894 q^{93} +22.6385 q^{95} -6.67478 q^{97} -1.17780 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	−2.04397	−1.18008	−0.590042	−	0.807372i	$-0.700889\pi$
$3$	−2.04397	−1.18008	−0.590042	+	0.807372i	$0.700889\pi$
$4$	0	0
$5$	4.25466	1.90274	0.951372	−	0.308045i	$-0.0996746\pi$
$5$	4.25466	1.90274	0.951372	+	0.308045i	$0.0996746\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.17780	0.392600
$10$	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
$15$	−8.69639	−2.24540
$16$	0	0
$17$	−2.35304	−0.570697	−0.285348	−	0.958424i	$-0.592109\pi$
$17$	−2.35304	−0.570697	−0.285348	+	0.958424i	$0.592109\pi$
$18$	0	0
$19$	5.32087	1.22069	0.610346	−	0.792135i	$-0.291031\pi$
$19$	5.32087	1.22069	0.610346	+	0.792135i	$0.291031\pi$
$20$	0	0
$21$	2.04397	0.446030
$22$	0	0
$23$	3.35219	0.698981	0.349490	−	0.936940i	$-0.386355\pi$
$23$	3.35219	0.698981	0.349490	+	0.936940i	$0.386355\pi$
$24$	0	0
$25$	13.1022	2.62043
$26$	0	0
$27$	3.72452	0.716784
$28$	0	0
$29$	−0.720205	−0.133739	−0.0668693	−	0.997762i	$-0.521301\pi$
$29$	−0.720205	−0.133739	−0.0668693	+	0.997762i	$0.521301\pi$
$30$	0	0
$31$	−9.19263	−1.65104	−0.825522	−	0.564369i	$-0.809119\pi$
$31$	−9.19263	−1.65104	−0.825522	+	0.564369i	$0.809119\pi$
$32$	0	0
$33$	2.04397	0.355809
$34$	0	0
$35$	−4.25466	−0.719170
$36$	0	0
$37$	7.52534	1.23716	0.618579	−	0.785723i	$-0.287709\pi$
$37$	7.52534	1.23716	0.618579	+	0.785723i	$0.287709\pi$
$38$	0	0
$39$	−2.04397	−0.327297
$40$	0	0
$41$	1.34603	0.210215	0.105108	−	0.994461i	$-0.466481\pi$
$41$	1.34603	0.210215	0.105108	+	0.994461i	$0.466481\pi$
$42$	0	0
$43$	−5.57303	−0.849879	−0.424940	−	0.905222i	$-0.639705\pi$
$43$	−5.57303	−0.849879	−0.424940	+	0.905222i	$0.639705\pi$
$44$	0	0
$45$	5.01114	0.747016
$46$	0	0
$47$	−5.81408	−0.848070	−0.424035	−	0.905646i	$-0.639387\pi$
$47$	−5.81408	−0.848070	−0.424035	+	0.905646i	$0.639387\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.80954	0.673470
$52$	0	0
$53$	3.10736	0.426829	0.213415	−	0.976962i	$-0.431541\pi$
$53$	3.10736	0.426829	0.213415	+	0.976962i	$0.431541\pi$
$54$	0	0
$55$	−4.25466	−0.573699
$56$	0	0
$57$	−10.8757	−1.44052
$58$	0	0
$59$	2.11450	0.275285	0.137642	−	0.990482i	$-0.456048\pi$
$59$	2.11450	0.275285	0.137642	+	0.990482i	$0.456048\pi$
$60$	0	0
$61$	9.70962	1.24319	0.621595	−	0.783339i	$-0.286485\pi$
$61$	9.70962	1.24319	0.621595	+	0.783339i	$0.286485\pi$
$62$	0	0
$63$	−1.17780	−0.148389
$64$	0	0
$65$	4.25466	0.527726
$66$	0	0
$67$	−5.48070	−0.669575	−0.334787	−	0.942294i	$-0.608664\pi$
$67$	−5.48070	−0.669575	−0.334787	+	0.942294i	$0.608664\pi$
$68$	0	0
$69$	−6.85177	−0.824857
$70$	0	0
$71$	−7.12854	−0.846003	−0.423001	−	0.906129i	$-0.639023\pi$
$71$	−7.12854	−0.846003	−0.423001	+	0.906129i	$0.639023\pi$
$72$	0	0
$73$	−0.258610	−0.0302681	−0.0151340	−	0.999885i	$-0.504817\pi$
$73$	−0.258610	−0.0302681	−0.0151340	+	0.999885i	$0.504817\pi$
$74$	0	0
$75$	−26.7804	−3.09233
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	6.57983	0.740288	0.370144	−	0.928974i	$-0.379308\pi$
$79$	6.57983	0.740288	0.370144	+	0.928974i	$0.379308\pi$
$80$	0	0
$81$	−11.1462	−1.23847
$82$	0	0
$83$	13.7748	1.51198	0.755991	−	0.654582i	$-0.227155\pi$
$83$	13.7748	1.51198	0.755991	+	0.654582i	$0.227155\pi$
$84$	0	0
$85$	−10.0114	−1.08589
$86$	0	0
$87$	1.47207	0.157823
$88$	0	0
$89$	−3.05729	−0.324072	−0.162036	−	0.986785i	$-0.551806\pi$
$89$	−3.05729	−0.324072	−0.162036	+	0.986785i	$0.551806\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	18.7894	1.94837
$94$	0	0
$95$	22.6385	2.32266
$96$	0	0
$97$	−6.67478	−0.677722	−0.338861	−	0.940837i	$-0.610042\pi$
$97$	−6.67478	−0.677722	−0.338861	+	0.940837i	$0.610042\pi$
$98$	0	0
$99$	−1.17780	−0.118373
$100$	0	0
$101$	1.35677	0.135003	0.0675017	−	0.997719i	$-0.478497\pi$
$101$	1.35677	0.135003	0.0675017	+	0.997719i	$0.478497\pi$
$102$	0	0
$103$	9.37969	0.924208	0.462104	−	0.886826i	$-0.347095\pi$
$103$	9.37969	0.924208	0.462104	+	0.886826i	$0.347095\pi$
$104$	0	0
$105$	8.69639	0.848681
$106$	0	0
$107$	6.08506	0.588265	0.294132	−	0.955765i	$-0.404969\pi$
$107$	6.08506	0.588265	0.294132	+	0.955765i	$0.404969\pi$
$108$	0	0
$109$	5.74046	0.549837	0.274918	−	0.961468i	$-0.411349\pi$
$109$	5.74046	0.549837	0.274918	+	0.961468i	$0.411349\pi$
$110$	0	0
$111$	−15.3815	−1.45995
$112$	0	0
$113$	16.7217	1.57305	0.786523	−	0.617561i	$-0.211879\pi$
$113$	16.7217	1.57305	0.786523	+	0.617561i	$0.211879\pi$
$114$	0	0
$115$	14.2625	1.32998
$116$	0	0
$117$	1.17780	0.108888
$118$	0	0
$119$	2.35304	0.215703
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.75125	−0.248072
$124$	0	0
$125$	34.4720	3.08327
$126$	0	0
$127$	−12.0817	−1.07208	−0.536040	−	0.844193i	$-0.680080\pi$
$127$	−12.0817	−1.07208	−0.536040	+	0.844193i	$0.680080\pi$
$128$	0	0
$129$	11.3911	1.00293
$130$	0	0
$131$	9.32641	0.814852	0.407426	−	0.913238i	$-0.366426\pi$
$131$	9.32641	0.814852	0.407426	+	0.913238i	$0.366426\pi$
$132$	0	0
$133$	−5.32087	−0.461378
$134$	0	0
$135$	15.8466	1.36386
$136$	0	0
$137$	−19.6463	−1.67849	−0.839247	−	0.543751i	$-0.817004\pi$
$137$	−19.6463	−1.67849	−0.839247	+	0.543751i	$0.817004\pi$
$138$	0	0
$139$	−17.6510	−1.49713	−0.748567	−	0.663059i	$-0.769258\pi$
$139$	−17.6510	−1.49713	−0.748567	+	0.663059i	$0.769258\pi$
$140$	0	0
$141$	11.8838	1.00079
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−3.06423	−0.254470
$146$	0	0
$147$	−2.04397	−0.168584
$148$	0	0
$149$	1.60263	0.131293	0.0656463	−	0.997843i	$-0.479089\pi$
$149$	1.60263	0.131293	0.0656463	+	0.997843i	$0.479089\pi$
$150$	0	0
$151$	−9.21254	−0.749706	−0.374853	−	0.927084i	$-0.622307\pi$
$151$	−9.21254	−0.749706	−0.374853	+	0.927084i	$0.622307\pi$
$152$	0	0
$153$	−2.77141	−0.224055
$154$	0	0
$155$	−39.1115	−3.14152
$156$	0	0
$157$	12.7064	1.01408	0.507041	−	0.861922i	$-0.330739\pi$
$157$	12.7064	1.01408	0.507041	+	0.861922i	$0.330739\pi$
$158$	0	0
$159$	−6.35135	−0.503695
$160$	0	0
$161$	−3.35219	−0.264190
$162$	0	0
$163$	−13.5036	−1.05769	−0.528843	−	0.848720i	$-0.677374\pi$
$163$	−13.5036	−1.05769	−0.528843	+	0.848720i	$0.677374\pi$
$164$	0	0
$165$	8.69639	0.677013
$166$	0	0
$167$	15.1604	1.17315	0.586574	−	0.809895i	$-0.300476\pi$
$167$	15.1604	1.17315	0.586574	+	0.809895i	$0.300476\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.26691	0.479243
$172$	0	0
$173$	−5.12675	−0.389780	−0.194890	−	0.980825i	$-0.562435\pi$
$173$	−5.12675	−0.389780	−0.194890	+	0.980825i	$0.562435\pi$
$174$	0	0
$175$	−13.1022	−0.990431
$176$	0	0
$177$	−4.32197	−0.324859
$178$	0	0
$179$	22.3917	1.67364	0.836818	−	0.547481i	$-0.184413\pi$
$179$	22.3917	1.67364	0.836818	+	0.547481i	$0.184413\pi$
$180$	0	0
$181$	23.4564	1.74350	0.871751	−	0.489948i	$-0.162984\pi$
$181$	23.4564	1.74350	0.871751	+	0.489948i	$0.162984\pi$
$182$	0	0
$183$	−19.8461	−1.46707
$184$	0	0
$185$	32.0178	2.35399
$186$	0	0
$187$	2.35304	0.172071
$188$	0	0
$189$	−3.72452	−0.270919
$190$	0	0
$191$	21.4771	1.55403	0.777014	−	0.629484i	$-0.216733\pi$
$191$	21.4771	1.55403	0.777014	+	0.629484i	$0.216733\pi$
$192$	0	0
$193$	−1.48006	−0.106537	−0.0532686	−	0.998580i	$-0.516964\pi$
$193$	−1.48006	−0.106537	−0.0532686	+	0.998580i	$0.516964\pi$
$194$	0	0
$195$	−8.69639	−0.622762
$196$	0	0
$197$	−6.37638	−0.454298	−0.227149	−	0.973860i	$-0.572941\pi$
$197$	−6.37638	−0.454298	−0.227149	+	0.973860i	$0.572941\pi$
$198$	0	0
$199$	−3.50808	−0.248681	−0.124341	−	0.992240i	$-0.539682\pi$
$199$	−3.50808	−0.248681	−0.124341	+	0.992240i	$0.539682\pi$
$200$	0	0
$201$	11.2024	0.790155
$202$	0	0
$203$	0.720205	0.0505485
$204$	0	0
$205$	5.72692	0.399986
$206$	0	0
$207$	3.94821	0.274420
$208$	0	0
$209$	−5.32087	−0.368052
$210$	0	0
$211$	20.1802	1.38926	0.694632	−	0.719365i	$-0.255567\pi$
$211$	20.1802	1.38926	0.694632	+	0.719365i	$0.255567\pi$
$212$	0	0
$213$	14.5705	0.998355
$214$	0	0
$215$	−23.7114	−1.61710
$216$	0	0
$217$	9.19263	0.624036
$218$	0	0
$219$	0.528591	0.0357189
$220$	0	0
$221$	−2.35304	−0.158283
$222$	0	0
$223$	−3.38320	−0.226556	−0.113278	−	0.993563i	$-0.536135\pi$
$223$	−3.38320	−0.226556	−0.113278	+	0.993563i	$0.536135\pi$
$224$	0	0
$225$	15.4317	1.02878
$226$	0	0
$227$	1.82607	0.121200	0.0606001	−	0.998162i	$-0.480699\pi$
$227$	1.82607	0.121200	0.0606001	+	0.998162i	$0.480699\pi$
$228$	0	0
$229$	29.9751	1.98081	0.990407	−	0.138183i	$-0.0441261\pi$
$229$	29.9751	1.98081	0.990407	+	0.138183i	$0.0441261\pi$
$230$	0	0
$231$	−2.04397	−0.134483
$232$	0	0
$233$	24.1806	1.58412	0.792062	−	0.610441i	$-0.209008\pi$
$233$	24.1806	1.58412	0.792062	+	0.610441i	$0.209008\pi$
$234$	0	0
$235$	−24.7369	−1.61366
$236$	0	0
$237$	−13.4489	−0.873603
$238$	0	0
$239$	−17.8296	−1.15330	−0.576650	−	0.816991i	$-0.695640\pi$
$239$	−17.8296	−1.15330	−0.576650	+	0.816991i	$0.695640\pi$
$240$	0	0
$241$	−14.3192	−0.922382	−0.461191	−	0.887301i	$-0.652578\pi$
$241$	−14.3192	−0.922382	−0.461191	+	0.887301i	$0.652578\pi$
$242$	0	0
$243$	11.6089	0.744710
$244$	0	0
$245$	4.25466	0.271821
$246$	0	0
$247$	5.32087	0.338559
$248$	0	0
$249$	−28.1553	−1.78427
$250$	0	0
$251$	−8.83024	−0.557360	−0.278680	−	0.960384i	$-0.589897\pi$
$251$	−8.83024	−0.557360	−0.278680	+	0.960384i	$0.589897\pi$
$252$	0	0
$253$	−3.35219	−0.210751
$254$	0	0
$255$	20.4630	1.28144
$256$	0	0
$257$	−14.9347	−0.931602	−0.465801	−	0.884890i	$-0.654234\pi$
$257$	−14.9347	−0.931602	−0.465801	+	0.884890i	$0.654234\pi$
$258$	0	0
$259$	−7.52534	−0.467602
$260$	0	0
$261$	−0.848256	−0.0525057
$262$	0	0
$263$	−7.67658	−0.473358	−0.236679	−	0.971588i	$-0.576059\pi$
$263$	−7.67658	−0.473358	−0.236679	+	0.971588i	$0.576059\pi$
$264$	0	0
$265$	13.2208	0.812147
$266$	0	0
$267$	6.24900	0.382433
$268$	0	0
$269$	10.0167	0.610729	0.305365	−	0.952236i	$-0.401222\pi$
$269$	10.0167	0.610729	0.305365	+	0.952236i	$0.401222\pi$
$270$	0	0
$271$	−20.1625	−1.22478	−0.612392	−	0.790555i	$-0.709792\pi$
$271$	−20.1625	−1.22478	−0.612392	+	0.790555i	$0.709792\pi$
$272$	0	0
$273$	2.04397	0.123706
$274$	0	0
$275$	−13.1022	−0.790091
$276$	0	0
$277$	−8.55993	−0.514316	−0.257158	−	0.966369i	$-0.582786\pi$
$277$	−8.55993	−0.514316	−0.257158	+	0.966369i	$0.582786\pi$
$278$	0	0
$279$	−10.8271	−0.648199
$280$	0	0
$281$	−16.8748	−1.00666	−0.503332	−	0.864093i	$-0.667893\pi$
$281$	−16.8748	−1.00666	−0.503332	+	0.864093i	$0.667893\pi$
$282$	0	0
$283$	5.35859	0.318535	0.159268	−	0.987235i	$-0.449087\pi$
$283$	5.35859	0.318535	0.159268	+	0.987235i	$0.449087\pi$
$284$	0	0
$285$	−46.2724	−2.74094
$286$	0	0
$287$	−1.34603	−0.0794539
$288$	0	0
$289$	−11.4632	−0.674305
$290$	0	0
$291$	13.6430	0.799769
$292$	0	0
$293$	−26.4402	−1.54466	−0.772328	−	0.635224i	$-0.780908\pi$
$293$	−26.4402	−1.54466	−0.772328	+	0.635224i	$0.780908\pi$
$294$	0	0
$295$	8.99649	0.523796
$296$	0	0
$297$	−3.72452	−0.216118
$298$	0	0
$299$	3.35219	0.193862
$300$	0	0
$301$	5.57303	0.321224
$302$	0	0
$303$	−2.77319	−0.159315
$304$	0	0
$305$	41.3112	2.36547
$306$	0	0
$307$	−4.53916	−0.259063	−0.129532	−	0.991575i	$-0.541347\pi$
$307$	−4.53916	−0.259063	−0.129532	+	0.991575i	$0.541347\pi$
$308$	0	0
$309$	−19.1718	−1.09064
$310$	0	0
$311$	−21.2870	−1.20707	−0.603537	−	0.797335i	$-0.706242\pi$
$311$	−21.2870	−1.20707	−0.603537	+	0.797335i	$0.706242\pi$
$312$	0	0
$313$	18.4052	1.04032	0.520162	−	0.854068i	$-0.325872\pi$
$313$	18.4052	1.04032	0.520162	+	0.854068i	$0.325872\pi$
$314$	0	0
$315$	−5.01114	−0.282346
$316$	0	0
$317$	−28.3848	−1.59425	−0.797125	−	0.603814i	$-0.793647\pi$
$317$	−28.3848	−1.59425	−0.797125	+	0.603814i	$0.793647\pi$
$318$	0	0
$319$	0.720205	0.0403237
$320$	0	0
$321$	−12.4377	−0.694202
$322$	0	0
$323$	−12.5202	−0.696644
$324$	0	0
$325$	13.1022	0.726778
$326$	0	0
$327$	−11.7333	−0.648854
$328$	0	0
$329$	5.81408	0.320540
$330$	0	0
$331$	15.3055	0.841268	0.420634	−	0.907230i	$-0.361808\pi$
$331$	15.3055	0.841268	0.420634	+	0.907230i	$0.361808\pi$
$332$	0	0
$333$	8.86333	0.485708
$334$	0	0
$335$	−23.3186	−1.27403
$336$	0	0
$337$	19.4956	1.06200	0.530998	−	0.847373i	$-0.321817\pi$
$337$	19.4956	1.06200	0.530998	+	0.847373i	$0.321817\pi$
$338$	0	0
$339$	−34.1786	−1.85633
$340$	0	0
$341$	9.19263	0.497809
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−29.1520	−1.56949
$346$	0	0
$347$	6.10894	0.327945	0.163972	−	0.986465i	$-0.447569\pi$
$347$	6.10894	0.327945	0.163972	+	0.986465i	$0.447569\pi$
$348$	0	0
$349$	12.1983	0.652958	0.326479	−	0.945204i	$-0.394138\pi$
$349$	12.1983	0.652958	0.326479	+	0.945204i	$0.394138\pi$
$350$	0	0
$351$	3.72452	0.198800
$352$	0	0
$353$	12.6076	0.671034	0.335517	−	0.942034i	$-0.391089\pi$
$353$	12.6076	0.671034	0.335517	+	0.942034i	$0.391089\pi$
$354$	0	0
$355$	−30.3296	−1.60973
$356$	0	0
$357$	−4.80954	−0.254548
$358$	0	0
$359$	17.2036	0.907974	0.453987	−	0.891008i	$-0.350001\pi$
$359$	17.2036	0.907974	0.453987	+	0.891008i	$0.350001\pi$
$360$	0	0
$361$	9.31165	0.490087
$362$	0	0
$363$	−2.04397	−0.107280
$364$	0	0
$365$	−1.10030	−0.0575924
$366$	0	0
$367$	26.3213	1.37396	0.686980	−	0.726676i	$-0.258936\pi$
$367$	26.3213	1.37396	0.686980	+	0.726676i	$0.258936\pi$
$368$	0	0
$369$	1.58536	0.0825304
$370$	0	0
$371$	−3.10736	−0.161326
$372$	0	0
$373$	−10.3855	−0.537740	−0.268870	−	0.963177i	$-0.586650\pi$
$373$	−10.3855	−0.537740	−0.268870	+	0.963177i	$0.586650\pi$
$374$	0	0
$375$	−70.4596	−3.63852
$376$	0	0
$377$	−0.720205	−0.0370924
$378$	0	0
$379$	6.17551	0.317215	0.158607	−	0.987342i	$-0.449300\pi$
$379$	6.17551	0.317215	0.158607	+	0.987342i	$0.449300\pi$
$380$	0	0
$381$	24.6946	1.26515
$382$	0	0
$383$	31.0723	1.58772	0.793859	−	0.608102i	$-0.208069\pi$
$383$	31.0723	1.58772	0.793859	+	0.608102i	$0.208069\pi$
$384$	0	0
$385$	4.25466	0.216838
$386$	0	0
$387$	−6.56391	−0.333662
$388$	0	0
$389$	23.5890	1.19601	0.598005	−	0.801492i	$-0.295960\pi$
$389$	23.5890	1.19601	0.598005	+	0.801492i	$0.295960\pi$
$390$	0	0
$391$	−7.88785	−0.398906
$392$	0	0
$393$	−19.0629	−0.961595
$394$	0	0
$395$	27.9950	1.40858
$396$	0	0
$397$	22.7356	1.14107	0.570533	−	0.821275i	$-0.306737\pi$
$397$	22.7356	1.14107	0.570533	+	0.821275i	$0.306737\pi$
$398$	0	0
$399$	10.8757	0.544465
$400$	0	0
$401$	−25.5650	−1.27665	−0.638326	−	0.769766i	$-0.720373\pi$
$401$	−25.5650	−1.27665	−0.638326	+	0.769766i	$0.720373\pi$
$402$	0	0
$403$	−9.19263	−0.457917
$404$	0	0
$405$	−47.4233	−2.35648
$406$	0	0
$407$	−7.52534	−0.373017
$408$	0	0
$409$	−24.1816	−1.19571	−0.597853	−	0.801606i	$-0.703979\pi$
$409$	−24.1816	−1.19571	−0.597853	+	0.801606i	$0.703979\pi$
$410$	0	0
$411$	40.1563	1.98076
$412$	0	0
$413$	−2.11450	−0.104048
$414$	0	0
$415$	58.6072	2.87691
$416$	0	0
$417$	36.0780	1.76674
$418$	0	0
$419$	1.41809	0.0692780	0.0346390	−	0.999400i	$-0.488972\pi$
$419$	1.41809	0.0692780	0.0346390	+	0.999400i	$0.488972\pi$
$420$	0	0
$421$	7.43848	0.362529	0.181265	−	0.983434i	$-0.441981\pi$
$421$	7.43848	0.362529	0.181265	+	0.983434i	$0.441981\pi$
$422$	0	0
$423$	−6.84781	−0.332952
$424$	0	0
$425$	−30.8300	−1.49547
$426$	0	0
$427$	−9.70962	−0.469881
$428$	0	0
$429$	2.04397	0.0986836
$430$	0	0
$431$	25.3898	1.22299	0.611493	−	0.791250i	$-0.290569\pi$
$431$	25.3898	1.22299	0.611493	+	0.791250i	$0.290569\pi$
$432$	0	0
$433$	21.6332	1.03962	0.519812	−	0.854281i	$-0.326002\pi$
$433$	21.6332	1.03962	0.519812	+	0.854281i	$0.326002\pi$
$434$	0	0
$435$	6.26318	0.300297
$436$	0	0
$437$	17.8366	0.853240
$438$	0	0
$439$	20.0140	0.955218	0.477609	−	0.878573i	$-0.341504\pi$
$439$	20.0140	0.955218	0.477609	+	0.878573i	$0.341504\pi$
$440$	0	0
$441$	1.17780	0.0560857
$442$	0	0
$443$	18.3744	0.872995	0.436497	−	0.899705i	$-0.356219\pi$
$443$	18.3744	0.872995	0.436497	+	0.899705i	$0.356219\pi$
$444$	0	0
$445$	−13.0078	−0.616627
$446$	0	0
$447$	−3.27572	−0.154936
$448$	0	0
$449$	15.2373	0.719093	0.359547	−	0.933127i	$-0.382931\pi$
$449$	15.2373	0.719093	0.359547	+	0.933127i	$0.382931\pi$
$450$	0	0
$451$	−1.34603	−0.0633822
$452$	0	0
$453$	18.8301	0.884716
$454$	0	0
$455$	−4.25466	−0.199462
$456$	0	0
$457$	20.9361	0.979350	0.489675	−	0.871905i	$-0.337115\pi$
$457$	20.9361	0.979350	0.489675	+	0.871905i	$0.337115\pi$
$458$	0	0
$459$	−8.76395	−0.409066
$460$	0	0
$461$	−22.3491	−1.04090	−0.520451	−	0.853891i	$-0.674236\pi$
$461$	−22.3491	−1.04090	−0.520451	+	0.853891i	$0.674236\pi$
$462$	0	0
$463$	−6.56648	−0.305170	−0.152585	−	0.988290i	$-0.548760\pi$
$463$	−6.56648	−0.305170	−0.152585	+	0.988290i	$0.548760\pi$
$464$	0	0
$465$	79.9427	3.70725
$466$	0	0
$467$	28.0743	1.29912	0.649561	−	0.760309i	$-0.274953\pi$
$467$	28.0743	1.29912	0.649561	+	0.760309i	$0.274953\pi$
$468$	0	0
$469$	5.48070	0.253075
$470$	0	0
$471$	−25.9715	−1.19670
$472$	0	0
$473$	5.57303	0.256248
$474$	0	0
$475$	69.7149	3.19874
$476$	0	0
$477$	3.65985	0.167573
$478$	0	0
$479$	−38.0758	−1.73973	−0.869864	−	0.493291i	$-0.835794\pi$
$479$	−38.0758	−1.73973	−0.869864	+	0.493291i	$0.835794\pi$
$480$	0	0
$481$	7.52534	0.343126
$482$	0	0
$483$	6.85177	0.311766
$484$	0	0
$485$	−28.3990	−1.28953
$486$	0	0
$487$	27.1870	1.23196	0.615979	−	0.787762i	$-0.288761\pi$
$487$	27.1870	1.23196	0.615979	+	0.787762i	$0.288761\pi$
$488$	0	0
$489$	27.6010	1.24816
$490$	0	0
$491$	14.0141	0.632449	0.316225	−	0.948684i	$-0.397585\pi$
$491$	14.0141	0.632449	0.316225	+	0.948684i	$0.397585\pi$
$492$	0	0
$493$	1.69467	0.0763242
$494$	0	0
$495$	−5.01114	−0.225234
$496$	0	0
$497$	7.12854	0.319759
$498$	0	0
$499$	18.3518	0.821541	0.410771	−	0.911739i	$-0.365260\pi$
$499$	18.3518	0.821541	0.410771	+	0.911739i	$0.365260\pi$
$500$	0	0
$501$	−30.9874	−1.38441
$502$	0	0
$503$	−8.21422	−0.366254	−0.183127	−	0.983089i	$-0.558622\pi$
$503$	−8.21422	−0.366254	−0.183127	+	0.983089i	$0.558622\pi$
$504$	0	0
$505$	5.77259	0.256877
$506$	0	0
$507$	−2.04397	−0.0907757
$508$	0	0
$509$	−17.5294	−0.776977	−0.388489	−	0.921453i	$-0.627003\pi$
$509$	−17.5294	−0.776977	−0.388489	+	0.921453i	$0.627003\pi$
$510$	0	0
$511$	0.258610	0.0114402
$512$	0	0
$513$	19.8177	0.874972
$514$	0	0
$515$	39.9074	1.75853
$516$	0	0
$517$	5.81408	0.255703
$518$	0	0
$519$	10.4789	0.459973
$520$	0	0
$521$	4.53159	0.198532	0.0992662	−	0.995061i	$-0.468350\pi$
$521$	4.53159	0.198532	0.0992662	+	0.995061i	$0.468350\pi$
$522$	0	0
$523$	−40.3441	−1.76413	−0.882063	−	0.471132i	$-0.843845\pi$
$523$	−40.3441	−1.76413	−0.882063	+	0.471132i	$0.843845\pi$
$524$	0	0
$525$	26.7804	1.16879
$526$	0	0
$527$	21.6306	0.942245
$528$	0	0
$529$	−11.7628	−0.511426
$530$	0	0
$531$	2.49046	0.108077
$532$	0	0
$533$	1.34603	0.0583032
$534$	0	0
$535$	25.8899	1.11932
$536$	0	0
$537$	−45.7679	−1.97503
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−25.7233	−1.10593	−0.552965	−	0.833204i	$-0.686504\pi$
$541$	−25.7233	−1.10593	−0.552965	+	0.833204i	$0.686504\pi$
$542$	0	0
$543$	−47.9442	−2.05748
$544$	0	0
$545$	24.4238	1.04620
$546$	0	0
$547$	35.6637	1.52487	0.762434	−	0.647066i	$-0.224004\pi$
$547$	35.6637	1.52487	0.762434	+	0.647066i	$0.224004\pi$
$548$	0	0
$549$	11.4360	0.488076
$550$	0	0
$551$	−3.83212	−0.163254
$552$	0	0
$553$	−6.57983	−0.279803
$554$	0	0
$555$	−65.4433	−2.77791
$556$	0	0
$557$	18.6296	0.789363	0.394682	−	0.918818i	$-0.370855\pi$
$557$	18.6296	0.789363	0.394682	+	0.918818i	$0.370855\pi$
$558$	0	0
$559$	−5.57303	−0.235714
$560$	0	0
$561$	−4.80954	−0.203059
$562$	0	0
$563$	38.3809	1.61756	0.808780	−	0.588111i	$-0.200128\pi$
$563$	38.3809	1.61756	0.808780	+	0.588111i	$0.200128\pi$
$564$	0	0
$565$	71.1452	2.99310
$566$	0	0
$567$	11.1462	0.468096
$568$	0	0
$569$	4.02816	0.168869	0.0844347	−	0.996429i	$-0.473092\pi$
$569$	4.02816	0.168869	0.0844347	+	0.996429i	$0.473092\pi$
$570$	0	0
$571$	31.0099	1.29772	0.648862	−	0.760906i	$-0.275245\pi$
$571$	31.0099	1.29772	0.648862	+	0.760906i	$0.275245\pi$
$572$	0	0
$573$	−43.8984	−1.83388
$574$	0	0
$575$	43.9210	1.83163
$576$	0	0
$577$	5.72564	0.238362	0.119181	−	0.992873i	$-0.461973\pi$
$577$	5.72564	0.238362	0.119181	+	0.992873i	$0.461973\pi$
$578$	0	0
$579$	3.02520	0.125723
$580$	0	0
$581$	−13.7748	−0.571476
$582$	0	0
$583$	−3.10736	−0.128694
$584$	0	0
$585$	5.01114	0.207185
$586$	0	0
$587$	30.7308	1.26840	0.634198	−	0.773171i	$-0.281330\pi$
$587$	30.7308	1.26840	0.634198	+	0.773171i	$0.281330\pi$
$588$	0	0
$589$	−48.9128	−2.01542
$590$	0	0
$591$	13.0331	0.536111
$592$	0	0
$593$	42.9751	1.76478	0.882388	−	0.470523i	$-0.155935\pi$
$593$	42.9751	1.76478	0.882388	+	0.470523i	$0.155935\pi$
$594$	0	0
$595$	10.0114	0.410428
$596$	0	0
$597$	7.17040	0.293465
$598$	0	0
$599$	−41.6264	−1.70081	−0.850403	−	0.526131i	$-0.823642\pi$
$599$	−41.6264	−1.70081	−0.850403	+	0.526131i	$0.823642\pi$
$600$	0	0
$601$	−4.74626	−0.193604	−0.0968021	−	0.995304i	$-0.530861\pi$
$601$	−4.74626	−0.193604	−0.0968021	+	0.995304i	$0.530861\pi$
$602$	0	0
$603$	−6.45517	−0.262875
$604$	0	0
$605$	4.25466	0.172977
$606$	0	0
$607$	6.16607	0.250273	0.125137	−	0.992140i	$-0.460063\pi$
$607$	6.16607	0.250273	0.125137	+	0.992140i	$0.460063\pi$
$608$	0	0
$609$	−1.47207	−0.0596515
$610$	0	0
$611$	−5.81408	−0.235212
$612$	0	0
$613$	11.9582	0.482986	0.241493	−	0.970403i	$-0.422363\pi$
$613$	11.9582	0.482986	0.241493	+	0.970403i	$0.422363\pi$
$614$	0	0
$615$	−11.7056	−0.472017
$616$	0	0
$617$	6.16087	0.248027	0.124014	−	0.992281i	$-0.460423\pi$
$617$	6.16087	0.248027	0.124014	+	0.992281i	$0.460423\pi$
$618$	0	0
$619$	24.5729	0.987667	0.493834	−	0.869556i	$-0.335595\pi$
$619$	24.5729	0.987667	0.493834	+	0.869556i	$0.335595\pi$
$620$	0	0
$621$	12.4853	0.501018
$622$	0	0
$623$	3.05729	0.122488
$624$	0	0
$625$	81.1560	3.24624
$626$	0	0
$627$	10.8757	0.434333
$628$	0	0
$629$	−17.7074	−0.706042
$630$	0	0
$631$	19.7080	0.784562	0.392281	−	0.919845i	$-0.371686\pi$
$631$	19.7080	0.784562	0.392281	+	0.919845i	$0.371686\pi$
$632$	0	0
$633$	−41.2477	−1.63945
$634$	0	0
$635$	−51.4037	−2.03989
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−8.39599	−0.332140
$640$	0	0
$641$	42.2023	1.66689	0.833445	−	0.552602i	$-0.186365\pi$
$641$	42.2023	1.66689	0.833445	+	0.552602i	$0.186365\pi$
$642$	0	0
$643$	3.39415	0.133852	0.0669262	−	0.997758i	$-0.478681\pi$
$643$	3.39415	0.133852	0.0669262	+	0.997758i	$0.478681\pi$
$644$	0	0
$645$	48.4653	1.90832
$646$	0	0
$647$	11.9597	0.470184	0.235092	−	0.971973i	$-0.424461\pi$
$647$	11.9597	0.470184	0.235092	+	0.971973i	$0.424461\pi$
$648$	0	0
$649$	−2.11450	−0.0830014
$650$	0	0
$651$	−18.7894	−0.736416
$652$	0	0
$653$	28.5304	1.11648	0.558241	−	0.829679i	$-0.311477\pi$
$653$	28.5304	1.11648	0.558241	+	0.829679i	$0.311477\pi$
$654$	0	0
$655$	39.6808	1.55046
$656$	0	0
$657$	−0.304591	−0.0118832
$658$	0	0
$659$	−5.49617	−0.214100	−0.107050	−	0.994254i	$-0.534141\pi$
$659$	−5.49617	−0.214100	−0.107050	+	0.994254i	$0.534141\pi$
$660$	0	0
$661$	35.6877	1.38809	0.694045	−	0.719931i	$-0.255827\pi$
$661$	35.6877	1.38809	0.694045	+	0.719931i	$0.255827\pi$
$662$	0	0
$663$	4.80954	0.186787
$664$	0	0
$665$	−22.6385	−0.877884
$666$	0	0
$667$	−2.41427	−0.0934808
$668$	0	0
$669$	6.91514	0.267355
$670$	0	0
$671$	−9.70962	−0.374836
$672$	0	0
$673$	20.7375	0.799372	0.399686	−	0.916652i	$-0.369119\pi$
$673$	20.7375	0.799372	0.399686	+	0.916652i	$0.369119\pi$
$674$	0	0
$675$	48.7993	1.87828
$676$	0	0
$677$	15.2675	0.586779	0.293390	−	0.955993i	$-0.405217\pi$
$677$	15.2675	0.586779	0.293390	+	0.955993i	$0.405217\pi$
$678$	0	0
$679$	6.67478	0.256155
$680$	0	0
$681$	−3.73242	−0.143027
$682$	0	0
$683$	−19.5772	−0.749100	−0.374550	−	0.927207i	$-0.622203\pi$
$683$	−19.5772	−0.749100	−0.374550	+	0.927207i	$0.622203\pi$
$684$	0	0
$685$	−83.5882	−3.19374
$686$	0	0
$687$	−61.2682	−2.33753
$688$	0	0
$689$	3.10736	0.118381
$690$	0	0
$691$	−26.5110	−1.00853	−0.504264	−	0.863550i	$-0.668236\pi$
$691$	−26.5110	−1.00853	−0.504264	+	0.863550i	$0.668236\pi$
$692$	0	0
$693$	1.17780	0.0447409
$694$	0	0
$695$	−75.0989	−2.84866
$696$	0	0
$697$	−3.16727	−0.119969
$698$	0	0
$699$	−49.4243	−1.86940
$700$	0	0
$701$	28.1777	1.06426	0.532128	−	0.846664i	$-0.321392\pi$
$701$	28.1777	1.06426	0.532128	+	0.846664i	$0.321392\pi$
$702$	0	0
$703$	40.0413	1.51019
$704$	0	0
$705$	50.5615	1.90426
$706$	0	0
$707$	−1.35677	−0.0510265
$708$	0	0
$709$	−15.1055	−0.567298	−0.283649	−	0.958928i	$-0.591545\pi$
$709$	−15.1055	−0.567298	−0.283649	+	0.958928i	$0.591545\pi$
$710$	0	0
$711$	7.74971	0.290637
$712$	0	0
$713$	−30.8155	−1.15405
$714$	0	0
$715$	−4.25466	−0.159115
$716$	0	0
$717$	36.4431	1.36099
$718$	0	0
$719$	−7.54607	−0.281421	−0.140710	−	0.990051i	$-0.544939\pi$
$719$	−7.54607	−0.281421	−0.140710	+	0.990051i	$0.544939\pi$
$720$	0	0
$721$	−9.37969	−0.349318
$722$	0	0
$723$	29.2680	1.08849
$724$	0	0
$725$	−9.43624	−0.350453
$726$	0	0
$727$	16.4012	0.608289	0.304144	−	0.952626i	$-0.401630\pi$
$727$	16.4012	0.608289	0.304144	+	0.952626i	$0.401630\pi$
$728$	0	0
$729$	9.71041	0.359645
$730$	0	0
$731$	13.1136	0.485023
$732$	0	0
$733$	−36.5200	−1.34890	−0.674449	−	0.738321i	$-0.735619\pi$
$733$	−36.5200	−1.34890	−0.674449	+	0.738321i	$0.735619\pi$
$734$	0	0
$735$	−8.69639	−0.320771
$736$	0	0
$737$	5.48070	0.201884
$738$	0	0
$739$	−12.6235	−0.464362	−0.232181	−	0.972673i	$-0.574586\pi$
$739$	−12.6235	−0.464362	−0.232181	+	0.972673i	$0.574586\pi$
$740$	0	0
$741$	−10.8757	−0.399528
$742$	0	0
$743$	−9.30740	−0.341455	−0.170728	−	0.985318i	$-0.554612\pi$
$743$	−9.30740	−0.341455	−0.170728	+	0.985318i	$0.554612\pi$
$744$	0	0
$745$	6.81866	0.249816
$746$	0	0
$747$	16.2240	0.593604
$748$	0	0
$749$	−6.08506	−0.222343
$750$	0	0
$751$	−20.6242	−0.752587	−0.376294	−	0.926500i	$-0.622802\pi$
$751$	−20.6242	−0.752587	−0.376294	+	0.926500i	$0.622802\pi$
$752$	0	0
$753$	18.0487	0.657732
$754$	0	0
$755$	−39.1963	−1.42650
$756$	0	0
$757$	−39.1060	−1.42133	−0.710665	−	0.703531i	$-0.751606\pi$
$757$	−39.1060	−1.42133	−0.710665	+	0.703531i	$0.751606\pi$
$758$	0	0
$759$	6.85177	0.248704
$760$	0	0
$761$	−51.7553	−1.87613	−0.938064	−	0.346463i	$-0.887383\pi$
$761$	−51.7553	−1.87613	−0.938064	+	0.346463i	$0.887383\pi$
$762$	0	0
$763$	−5.74046	−0.207819
$764$	0	0
$765$	−11.7914	−0.426320
$766$	0	0
$767$	2.11450	0.0763502
$768$	0	0
$769$	−16.5671	−0.597426	−0.298713	−	0.954343i	$-0.596557\pi$
$769$	−16.5671	−0.597426	−0.298713	+	0.954343i	$0.596557\pi$
$770$	0	0
$771$	30.5261	1.09937
$772$	0	0
$773$	−16.2905	−0.585928	−0.292964	−	0.956123i	$-0.594642\pi$
$773$	−16.2905	−0.585928	−0.292964	+	0.956123i	$0.594642\pi$
$774$	0	0
$775$	−120.443	−4.32645
$776$	0	0
$777$	15.3815	0.551809
$778$	0	0
$779$	7.16207	0.256608
$780$	0	0
$781$	7.12854	0.255079
$782$	0	0
$783$	−2.68242	−0.0958617
$784$	0	0
$785$	54.0615	1.92954
$786$	0	0
$787$	27.1269	0.966969	0.483484	−	0.875353i	$-0.339371\pi$
$787$	27.1269	0.966969	0.483484	+	0.875353i	$0.339371\pi$
$788$	0	0
$789$	15.6907	0.558603
$790$	0	0
$791$	−16.7217	−0.594555
$792$	0	0
$793$	9.70962	0.344799
$794$	0	0
$795$	−27.0229	−0.958402
$796$	0	0
$797$	−12.5297	−0.443826	−0.221913	−	0.975067i	$-0.571230\pi$
$797$	−12.5297	−0.443826	−0.221913	+	0.975067i	$0.571230\pi$
$798$	0	0
$799$	13.6808	0.483991
$800$	0	0
$801$	−3.60087	−0.127231
$802$	0	0
$803$	0.258610	0.00912616
$804$	0	0
$805$	−14.2625	−0.502686
$806$	0	0
$807$	−20.4738	−0.720712
$808$	0	0
$809$	−6.51409	−0.229023	−0.114512	−	0.993422i	$-0.536530\pi$
$809$	−6.51409	−0.229023	−0.114512	+	0.993422i	$0.536530\pi$
$810$	0	0
$811$	−11.1838	−0.392717	−0.196358	−	0.980532i	$-0.562912\pi$
$811$	−11.1838	−0.392717	−0.196358	+	0.980532i	$0.562912\pi$
$812$	0	0
$813$	41.2114	1.44535
$814$	0	0
$815$	−57.4534	−2.01251
$816$	0	0
$817$	−29.6534	−1.03744
$818$	0	0
$819$	−1.17780	−0.0411556
$820$	0	0
$821$	2.88261	0.100604	0.0503019	−	0.998734i	$-0.483982\pi$
$821$	2.88261	0.100604	0.0503019	+	0.998734i	$0.483982\pi$
$822$	0	0
$823$	40.4628	1.41045	0.705223	−	0.708986i	$-0.250847\pi$
$823$	40.4628	1.41045	0.705223	+	0.708986i	$0.250847\pi$
$824$	0	0
$825$	26.7804	0.932374
$826$	0	0
$827$	40.5984	1.41174	0.705872	−	0.708339i	$-0.250555\pi$
$827$	40.5984	1.41174	0.705872	+	0.708339i	$0.250555\pi$
$828$	0	0
$829$	−10.7496	−0.373349	−0.186674	−	0.982422i	$-0.559771\pi$
$829$	−10.7496	−0.373349	−0.186674	+	0.982422i	$0.559771\pi$
$830$	0	0
$831$	17.4962	0.606937
$832$	0	0
$833$	−2.35304	−0.0815281
$834$	0	0
$835$	64.5025	2.23220
$836$	0	0
$837$	−34.2381	−1.18344
$838$	0	0
$839$	45.8016	1.58125	0.790624	−	0.612303i	$-0.209757\pi$
$839$	45.8016	1.58125	0.790624	+	0.612303i	$0.209757\pi$
$840$	0	0
$841$	−28.4813	−0.982114
$842$	0	0
$843$	34.4915	1.18795
$844$	0	0
$845$	4.25466	0.146365
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−10.9528	−0.375899
$850$	0	0
$851$	25.2264	0.864749
$852$	0	0
$853$	−16.5317	−0.566035	−0.283018	−	0.959115i	$-0.591336\pi$
$853$	−16.5317	−0.566035	−0.283018	+	0.959115i	$0.591336\pi$
$854$	0	0
$855$	26.6636	0.911876
$856$	0	0
$857$	−25.7692	−0.880259	−0.440129	−	0.897934i	$-0.645067\pi$
$857$	−25.7692	−0.880259	−0.440129	+	0.897934i	$0.645067\pi$
$858$	0	0
$859$	−27.6228	−0.942479	−0.471239	−	0.882005i	$-0.656193\pi$
$859$	−27.6228	−0.942479	−0.471239	+	0.882005i	$0.656193\pi$
$860$	0	0
$861$	2.75125	0.0937623
$862$	0	0
$863$	52.7060	1.79413	0.897066	−	0.441896i	$-0.145694\pi$
$863$	52.7060	1.79413	0.897066	+	0.441896i	$0.145694\pi$
$864$	0	0
$865$	−21.8126	−0.741651
$866$	0	0
$867$	23.4304	0.795738
$868$	0	0
$869$	−6.57983	−0.223205
$870$	0	0
$871$	−5.48070	−0.185707
$872$	0	0
$873$	−7.86155	−0.266073
$874$	0	0
$875$	−34.4720	−1.16537
$876$	0	0
$877$	−10.9075	−0.368320	−0.184160	−	0.982896i	$-0.558956\pi$
$877$	−10.9075	−0.368320	−0.184160	+	0.982896i	$0.558956\pi$
$878$	0	0
$879$	54.0430	1.82282
$880$	0	0
$881$	−38.7056	−1.30402	−0.652012	−	0.758209i	$-0.726075\pi$
$881$	−38.7056	−1.30402	−0.652012	+	0.758209i	$0.726075\pi$
$882$	0	0
$883$	−34.1567	−1.14946	−0.574732	−	0.818342i	$-0.694894\pi$
$883$	−34.1567	−1.14946	−0.574732	+	0.818342i	$0.694894\pi$
$884$	0	0
$885$	−18.3885	−0.618123
$886$	0	0
$887$	−56.9048	−1.91068	−0.955338	−	0.295515i	$-0.904509\pi$
$887$	−56.9048	−1.91068	−0.955338	+	0.295515i	$0.904509\pi$
$888$	0	0
$889$	12.0817	0.405208
$890$	0	0
$891$	11.1462	0.373411
$892$	0	0
$893$	−30.9359	−1.03523
$894$	0	0
$895$	95.2693	3.18450
$896$	0	0
$897$	−6.85177	−0.228774
$898$	0	0
$899$	6.62057	0.220809
$900$	0	0
$901$	−7.31176	−0.243590
$902$	0	0
$903$	−11.3911	−0.379072
$904$	0	0
$905$	99.7992	3.31744
$906$	0	0
$907$	59.3467	1.97058	0.985288	−	0.170902i	$-0.0546682\pi$
$907$	59.3467	1.97058	0.985288	+	0.170902i	$0.0546682\pi$
$908$	0	0
$909$	1.59800	0.0530023
$910$	0	0
$911$	−30.6951	−1.01697	−0.508487	−	0.861069i	$-0.669795\pi$
$911$	−30.6951	−1.01697	−0.508487	+	0.861069i	$0.669795\pi$
$912$	0	0
$913$	−13.7748	−0.455880
$914$	0	0
$915$	−84.4386	−2.79146
$916$	0	0
$917$	−9.32641	−0.307985
$918$	0	0
$919$	6.25510	0.206337	0.103168	−	0.994664i	$-0.467102\pi$
$919$	6.25510	0.206337	0.103168	+	0.994664i	$0.467102\pi$
$920$	0	0
$921$	9.27788	0.305717
$922$	0	0
$923$	−7.12854	−0.234639
$924$	0	0
$925$	98.5982	3.24189
$926$	0	0
$927$	11.0474	0.362844
$928$	0	0
$929$	−31.1670	−1.02255	−0.511277	−	0.859416i	$-0.670827\pi$
$929$	−31.1670	−1.02255	−0.511277	+	0.859416i	$0.670827\pi$
$930$	0	0
$931$	5.32087	0.174384
$932$	0	0
$933$	43.5098	1.42445
$934$	0	0
$935$	10.0114	0.327408
$936$	0	0
$937$	−31.8319	−1.03990	−0.519951	−	0.854196i	$-0.674050\pi$
$937$	−31.8319	−1.03990	−0.519951	+	0.854196i	$0.674050\pi$
$938$	0	0
$939$	−37.6196	−1.22767
$940$	0	0
$941$	−52.1604	−1.70038	−0.850190	−	0.526476i	$-0.823513\pi$
$941$	−52.1604	−1.70038	−0.850190	+	0.526476i	$0.823513\pi$
$942$	0	0
$943$	4.51217	0.146936
$944$	0	0
$945$	−15.8466	−0.515489
$946$	0	0
$947$	−25.2096	−0.819203	−0.409602	−	0.912264i	$-0.634332\pi$
$947$	−25.2096	−0.819203	−0.409602	+	0.912264i	$0.634332\pi$
$948$	0	0
$949$	−0.258610	−0.00839485
$950$	0	0
$951$	58.0176	1.88135
$952$	0	0
$953$	26.6030	0.861757	0.430878	−	0.902410i	$-0.358204\pi$
$953$	26.6030	0.861757	0.430878	+	0.902410i	$0.358204\pi$
$954$	0	0
$955$	91.3778	2.95692
$956$	0	0
$957$	−1.47207	−0.0475854
$958$	0	0
$959$	19.6463	0.634411
$960$	0	0
$961$	53.5044	1.72595
$962$	0	0
$963$	7.16697	0.230952
$964$	0	0
$965$	−6.29717	−0.202713
$966$	0	0
$967$	27.5404	0.885641	0.442821	−	0.896610i	$-0.353978\pi$
$967$	27.5404	0.885641	0.442821	+	0.896610i	$0.353978\pi$
$968$	0	0
$969$	25.5909	0.822099
$970$	0	0
$971$	−1.11187	−0.0356817	−0.0178408	−	0.999841i	$-0.505679\pi$
$971$	−1.11187	−0.0356817	−0.0178408	+	0.999841i	$0.505679\pi$
$972$	0	0
$973$	17.6510	0.565864
$974$	0	0
$975$	−26.7804	−0.857659
$976$	0	0
$977$	4.41320	0.141191	0.0705953	−	0.997505i	$-0.477510\pi$
$977$	4.41320	0.141191	0.0705953	+	0.997505i	$0.477510\pi$
$978$	0	0
$979$	3.05729	0.0977115
$980$	0	0
$981$	6.76111	0.215866
$982$	0	0
$983$	−0.520610	−0.0166049	−0.00830245	−	0.999966i	$-0.502643\pi$
$983$	−0.520610	−0.0166049	−0.00830245	+	0.999966i	$0.502643\pi$
$984$	0	0
$985$	−27.1294	−0.864414
$986$	0	0
$987$	−11.8838	−0.378265
$988$	0	0
$989$	−18.6819	−0.594049
$990$	0	0
$991$	14.3800	0.456795	0.228397	−	0.973568i	$-0.426651\pi$
$991$	14.3800	0.456795	0.228397	+	0.973568i	$0.426651\pi$
$992$	0	0
$993$	−31.2840	−0.992768
$994$	0	0
$995$	−14.9257	−0.473176
$996$	0	0
$997$	−16.8253	−0.532863	−0.266432	−	0.963854i	$-0.585845\pi$
$997$	−16.8253	−0.532863	−0.266432	+	0.963854i	$0.585845\pi$
$998$	0	0
$999$	28.0283	0.886775

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.4	✓	15	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.04397 q^{3} +4.25466 q^{5} -1.00000 q^{7} +1.17780 q^{9} +O(q^{10})\)	\(q-2.04397 q^{3} +4.25466 q^{5} -1.00000 q^{7} +1.17780 q^{9} -1.00000 q^{11} +1.00000 q^{13} -8.69639 q^{15} -2.35304 q^{17} +5.32087 q^{19} +2.04397 q^{21} +3.35219 q^{23} +13.1022 q^{25} +3.72452 q^{27} -0.720205 q^{29} -9.19263 q^{31} +2.04397 q^{33} -4.25466 q^{35} +7.52534 q^{37} -2.04397 q^{39} +1.34603 q^{41} -5.57303 q^{43} +5.01114 q^{45} -5.81408 q^{47} +1.00000 q^{49} +4.80954 q^{51} +3.10736 q^{53} -4.25466 q^{55} -10.8757 q^{57} +2.11450 q^{59} +9.70962 q^{61} -1.17780 q^{63} +4.25466 q^{65} -5.48070 q^{67} -6.85177 q^{69} -7.12854 q^{71} -0.258610 q^{73} -26.7804 q^{75} +1.00000 q^{77} +6.57983 q^{79} -11.1462 q^{81} +13.7748 q^{83} -10.0114 q^{85} +1.47207 q^{87} -3.05729 q^{89} -1.00000 q^{91} +18.7894 q^{93} +22.6385 q^{95} -6.67478 q^{97} -1.17780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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