LMFDB - Embedded newform 8016.2.a.bf.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$3.59055$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.59055 q^{5} -4.10828 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.59055 q^{5} -4.10828 q^{7} +1.00000 q^{9} -1.62131 q^{11} +5.55250 q^{13} -3.59055 q^{15} -5.51714 q^{17} +0.688902 q^{19} +4.10828 q^{21} +4.57662 q^{23} +7.89208 q^{25} -1.00000 q^{27} +3.57239 q^{29} -7.18724 q^{31} +1.62131 q^{33} -14.7510 q^{35} -10.0764 q^{37} -5.55250 q^{39} -9.17469 q^{41} -2.89063 q^{43} +3.59055 q^{45} +5.70322 q^{47} +9.87799 q^{49} +5.51714 q^{51} -0.787842 q^{53} -5.82139 q^{55} -0.688902 q^{57} +2.75855 q^{59} -1.60053 q^{61} -4.10828 q^{63} +19.9365 q^{65} -7.32502 q^{67} -4.57662 q^{69} +6.98319 q^{71} +0.181729 q^{73} -7.89208 q^{75} +6.66079 q^{77} +13.9222 q^{79} +1.00000 q^{81} -3.82200 q^{83} -19.8096 q^{85} -3.57239 q^{87} -6.78217 q^{89} -22.8112 q^{91} +7.18724 q^{93} +2.47354 q^{95} -7.41920 q^{97} -1.62131 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9 - q^11 + 8 * q^13 - 4 * q^15 + 3 * q^17 - 12 * q^19 + 11 * q^21 - 7 * q^23 + 18 * q^25 - 12 * q^27 + 5 * q^29 - 33 * q^31 + q^33 - 15 * q^35 + 8 * q^37 - 8 * q^39 - 6 * q^41 - 16 * q^43 + 4 * q^45 - 18 * q^47 + 25 * q^49 - 3 * q^51 + 20 * q^53 - 39 * q^55 + 12 * q^57 - 4 * q^59 + 10 * q^61 - 11 * q^63 - 9 * q^67 + 7 * q^69 - 11 * q^71 + 22 * q^73 - 18 * q^75 + 24 * q^77 - 56 * q^79 + 12 * q^81 - 26 * q^83 + 15 * q^85 - 5 * q^87 - 15 * q^89 - 11 * q^91 + 33 * q^93 - 3 * q^95 + 8 * q^97 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.59055	1.60574	0.802872	−	0.596151i	$-0.203304\pi$
$5$	3.59055	1.60574	0.802872	+	0.596151i	$0.203304\pi$
$6$	0	0
$7$	−4.10828	−1.55279	−0.776393	−	0.630250i	$-0.782953\pi$
$7$	−4.10828	−1.55279	−0.776393	+	0.630250i	$0.782953\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.62131	−0.488842	−0.244421	−	0.969669i	$-0.578598\pi$
$11$	−1.62131	−0.488842	−0.244421	+	0.969669i	$0.578598\pi$
$12$	0	0
$13$	5.55250	1.53999	0.769993	−	0.638052i	$-0.220260\pi$
$13$	5.55250	1.53999	0.769993	+	0.638052i	$0.220260\pi$
$14$	0	0
$15$	−3.59055	−0.927077
$16$	0	0
$17$	−5.51714	−1.33810	−0.669051	−	0.743216i	$-0.733299\pi$
$17$	−5.51714	−1.33810	−0.669051	+	0.743216i	$0.733299\pi$
$18$	0	0
$19$	0.688902	0.158045	0.0790225	−	0.996873i	$-0.474820\pi$
$19$	0.688902	0.158045	0.0790225	+	0.996873i	$0.474820\pi$
$20$	0	0
$21$	4.10828	0.896501
$22$	0	0
$23$	4.57662	0.954291	0.477145	−	0.878824i	$-0.341671\pi$
$23$	4.57662	0.954291	0.477145	+	0.878824i	$0.341671\pi$
$24$	0	0
$25$	7.89208	1.57842
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.57239	0.663376	0.331688	−	0.943389i	$-0.392382\pi$
$29$	3.57239	0.663376	0.331688	+	0.943389i	$0.392382\pi$
$30$	0	0
$31$	−7.18724	−1.29087	−0.645433	−	0.763817i	$-0.723323\pi$
$31$	−7.18724	−1.29087	−0.645433	+	0.763817i	$0.723323\pi$
$32$	0	0
$33$	1.62131	0.282233
$34$	0	0
$35$	−14.7510	−2.49338
$36$	0	0
$37$	−10.0764	−1.65655	−0.828273	−	0.560325i	$-0.810676\pi$
$37$	−10.0764	−1.65655	−0.828273	+	0.560325i	$0.810676\pi$
$38$	0	0
$39$	−5.55250	−0.889111
$40$	0	0
$41$	−9.17469	−1.43285	−0.716423	−	0.697666i	$-0.754222\pi$
$41$	−9.17469	−1.43285	−0.716423	+	0.697666i	$0.754222\pi$
$42$	0	0
$43$	−2.89063	−0.440817	−0.220408	−	0.975408i	$-0.570739\pi$
$43$	−2.89063	−0.440817	−0.220408	+	0.975408i	$0.570739\pi$
$44$	0	0
$45$	3.59055	0.535248
$46$	0	0
$47$	5.70322	0.831901	0.415950	−	0.909387i	$-0.363449\pi$
$47$	5.70322	0.831901	0.415950	+	0.909387i	$0.363449\pi$
$48$	0	0
$49$	9.87799	1.41114
$50$	0	0
$51$	5.51714	0.772554
$52$	0	0
$53$	−0.787842	−0.108218	−0.0541092	−	0.998535i	$-0.517232\pi$
$53$	−0.787842	−0.108218	−0.0541092	+	0.998535i	$0.517232\pi$
$54$	0	0
$55$	−5.82139	−0.784956
$56$	0	0
$57$	−0.688902	−0.0912473
$58$	0	0
$59$	2.75855	0.359132	0.179566	−	0.983746i	$-0.442531\pi$
$59$	2.75855	0.359132	0.179566	+	0.983746i	$0.442531\pi$
$60$	0	0
$61$	−1.60053	−0.204927	−0.102464	−	0.994737i	$-0.532673\pi$
$61$	−1.60053	−0.204927	−0.102464	+	0.994737i	$0.532673\pi$
$62$	0	0
$63$	−4.10828	−0.517595
$64$	0	0
$65$	19.9365	2.47282
$66$	0	0
$67$	−7.32502	−0.894893	−0.447447	−	0.894311i	$-0.647667\pi$
$67$	−7.32502	−0.894893	−0.447447	+	0.894311i	$0.647667\pi$
$68$	0	0
$69$	−4.57662	−0.550960
$70$	0	0
$71$	6.98319	0.828752	0.414376	−	0.910106i	$-0.364000\pi$
$71$	6.98319	0.828752	0.414376	+	0.910106i	$0.364000\pi$
$72$	0	0
$73$	0.181729	0.0212698	0.0106349	−	0.999943i	$-0.496615\pi$
$73$	0.181729	0.0212698	0.0106349	+	0.999943i	$0.496615\pi$
$74$	0	0
$75$	−7.89208	−0.911299
$76$	0	0
$77$	6.66079	0.759067
$78$	0	0
$79$	13.9222	1.56637	0.783184	−	0.621790i	$-0.213594\pi$
$79$	13.9222	1.56637	0.783184	+	0.621790i	$0.213594\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.82200	−0.419519	−0.209759	−	0.977753i	$-0.567268\pi$
$83$	−3.82200	−0.419519	−0.209759	+	0.977753i	$0.567268\pi$
$84$	0	0
$85$	−19.8096	−2.14865
$86$	0	0
$87$	−3.57239	−0.383000
$88$	0	0
$89$	−6.78217	−0.718908	−0.359454	−	0.933163i	$-0.617037\pi$
$89$	−6.78217	−0.718908	−0.359454	+	0.933163i	$0.617037\pi$
$90$	0	0
$91$	−22.8112	−2.39127
$92$	0	0
$93$	7.18724	0.745282
$94$	0	0
$95$	2.47354	0.253780
$96$	0	0
$97$	−7.41920	−0.753306	−0.376653	−	0.926354i	$-0.622925\pi$
$97$	−7.41920	−0.753306	−0.376653	+	0.926354i	$0.622925\pi$
$98$	0	0
$99$	−1.62131	−0.162947
$100$	0	0
$101$	−19.3734	−1.92773	−0.963863	−	0.266399i	$-0.914166\pi$
$101$	−19.3734	−1.92773	−0.963863	+	0.266399i	$0.914166\pi$
$102$	0	0
$103$	2.04851	0.201846	0.100923	−	0.994894i	$-0.467820\pi$
$103$	2.04851	0.201846	0.100923	+	0.994894i	$0.467820\pi$
$104$	0	0
$105$	14.7510	1.43955
$106$	0	0
$107$	−0.805768	−0.0778965	−0.0389482	−	0.999241i	$-0.512401\pi$
$107$	−0.805768	−0.0778965	−0.0389482	+	0.999241i	$0.512401\pi$
$108$	0	0
$109$	8.62091	0.825733	0.412867	−	0.910791i	$-0.364528\pi$
$109$	8.62091	0.825733	0.412867	+	0.910791i	$0.364528\pi$
$110$	0	0
$111$	10.0764	0.956407
$112$	0	0
$113$	4.93599	0.464339	0.232169	−	0.972675i	$-0.425418\pi$
$113$	4.93599	0.464339	0.232169	+	0.972675i	$0.425418\pi$
$114$	0	0
$115$	16.4326	1.53235
$116$	0	0
$117$	5.55250	0.513329
$118$	0	0
$119$	22.6660	2.07778
$120$	0	0
$121$	−8.37136	−0.761033
$122$	0	0
$123$	9.17469	0.827254
$124$	0	0
$125$	10.3842	0.928787
$126$	0	0
$127$	12.3106	1.09239	0.546196	−	0.837658i	$-0.316075\pi$
$127$	12.3106	1.09239	0.546196	+	0.837658i	$0.316075\pi$
$128$	0	0
$129$	2.89063	0.254506
$130$	0	0
$131$	−16.4098	−1.43373	−0.716867	−	0.697210i	$-0.754424\pi$
$131$	−16.4098	−1.43373	−0.716867	+	0.697210i	$0.754424\pi$
$132$	0	0
$133$	−2.83021	−0.245410
$134$	0	0
$135$	−3.59055	−0.309026
$136$	0	0
$137$	−16.2690	−1.38996	−0.694979	−	0.719030i	$-0.744586\pi$
$137$	−16.2690	−1.38996	−0.694979	+	0.719030i	$0.744586\pi$
$138$	0	0
$139$	5.33754	0.452724	0.226362	−	0.974043i	$-0.427317\pi$
$139$	5.33754	0.452724	0.226362	+	0.974043i	$0.427317\pi$
$140$	0	0
$141$	−5.70322	−0.480298
$142$	0	0
$143$	−9.00230	−0.752810
$144$	0	0
$145$	12.8269	1.06521
$146$	0	0
$147$	−9.87799	−0.814723
$148$	0	0
$149$	−9.27063	−0.759480	−0.379740	−	0.925093i	$-0.623986\pi$
$149$	−9.27063	−0.759480	−0.379740	+	0.925093i	$0.623986\pi$
$150$	0	0
$151$	20.4894	1.66741	0.833703	−	0.552213i	$-0.186216\pi$
$151$	20.4894	1.66741	0.833703	+	0.552213i	$0.186216\pi$
$152$	0	0
$153$	−5.51714	−0.446034
$154$	0	0
$155$	−25.8062	−2.07280
$156$	0	0
$157$	−3.90511	−0.311662	−0.155831	−	0.987784i	$-0.549805\pi$
$157$	−3.90511	−0.311662	−0.155831	+	0.987784i	$0.549805\pi$
$158$	0	0
$159$	0.787842	0.0624799
$160$	0	0
$161$	−18.8020	−1.48181
$162$	0	0
$163$	13.1543	1.03032	0.515161	−	0.857094i	$-0.327732\pi$
$163$	13.1543	1.03032	0.515161	+	0.857094i	$0.327732\pi$
$164$	0	0
$165$	5.82139	0.453195
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	17.8302	1.37156
$170$	0	0
$171$	0.688902	0.0526817
$172$	0	0
$173$	11.4082	0.867351	0.433676	−	0.901069i	$-0.357216\pi$
$173$	11.4082	0.867351	0.433676	+	0.901069i	$0.357216\pi$
$174$	0	0
$175$	−32.4229	−2.45094
$176$	0	0
$177$	−2.75855	−0.207345
$178$	0	0
$179$	−19.7878	−1.47901	−0.739503	−	0.673153i	$-0.764940\pi$
$179$	−19.7878	−1.47901	−0.739503	+	0.673153i	$0.764940\pi$
$180$	0	0
$181$	5.54242	0.411965	0.205982	−	0.978556i	$-0.433961\pi$
$181$	5.54242	0.411965	0.205982	+	0.978556i	$0.433961\pi$
$182$	0	0
$183$	1.60053	0.118315
$184$	0	0
$185$	−36.1798	−2.65999
$186$	0	0
$187$	8.94497	0.654121
$188$	0	0
$189$	4.10828	0.298834
$190$	0	0
$191$	−17.3823	−1.25774	−0.628871	−	0.777510i	$-0.716483\pi$
$191$	−17.3823	−1.25774	−0.628871	+	0.777510i	$0.716483\pi$
$192$	0	0
$193$	−10.1587	−0.731239	−0.365619	−	0.930765i	$-0.619143\pi$
$193$	−10.1587	−0.731239	−0.365619	+	0.930765i	$0.619143\pi$
$194$	0	0
$195$	−19.9365	−1.42769
$196$	0	0
$197$	12.6909	0.904193	0.452096	−	0.891969i	$-0.350676\pi$
$197$	12.6909	0.904193	0.452096	+	0.891969i	$0.350676\pi$
$198$	0	0
$199$	−27.3399	−1.93808	−0.969038	−	0.246913i	$-0.920584\pi$
$199$	−27.3399	−1.93808	−0.969038	+	0.246913i	$0.920584\pi$
$200$	0	0
$201$	7.32502	0.516667
$202$	0	0
$203$	−14.6764	−1.03008
$204$	0	0
$205$	−32.9422	−2.30078
$206$	0	0
$207$	4.57662	0.318097
$208$	0	0
$209$	−1.11692	−0.0772591
$210$	0	0
$211$	−17.6413	−1.21448	−0.607239	−	0.794519i	$-0.707723\pi$
$211$	−17.6413	−1.21448	−0.607239	+	0.794519i	$0.707723\pi$
$212$	0	0
$213$	−6.98319	−0.478480
$214$	0	0
$215$	−10.3790	−0.707839
$216$	0	0
$217$	29.5272	2.00444
$218$	0	0
$219$	−0.181729	−0.0122801
$220$	0	0
$221$	−30.6339	−2.06066
$222$	0	0
$223$	−20.6260	−1.38122	−0.690608	−	0.723229i	$-0.742657\pi$
$223$	−20.6260	−1.38122	−0.690608	+	0.723229i	$0.742657\pi$
$224$	0	0
$225$	7.89208	0.526138
$226$	0	0
$227$	6.16425	0.409135	0.204568	−	0.978852i	$-0.434421\pi$
$227$	6.16425	0.409135	0.204568	+	0.978852i	$0.434421\pi$
$228$	0	0
$229$	−19.4392	−1.28458	−0.642289	−	0.766462i	$-0.722015\pi$
$229$	−19.4392	−1.28458	−0.642289	+	0.766462i	$0.722015\pi$
$230$	0	0
$231$	−6.66079	−0.438248
$232$	0	0
$233$	19.8065	1.29757	0.648783	−	0.760973i	$-0.275278\pi$
$233$	19.8065	1.29757	0.648783	+	0.760973i	$0.275278\pi$
$234$	0	0
$235$	20.4777	1.33582
$236$	0	0
$237$	−13.9222	−0.904343
$238$	0	0
$239$	−1.11360	−0.0720327	−0.0360164	−	0.999351i	$-0.511467\pi$
$239$	−1.11360	−0.0720327	−0.0360164	+	0.999351i	$0.511467\pi$
$240$	0	0
$241$	−26.4165	−1.70164	−0.850820	−	0.525458i	$-0.823894\pi$
$241$	−26.4165	−1.70164	−0.850820	+	0.525458i	$0.823894\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	35.4675	2.26593
$246$	0	0
$247$	3.82513	0.243387
$248$	0	0
$249$	3.82200	0.242209
$250$	0	0
$251$	−1.81268	−0.114415	−0.0572077	−	0.998362i	$-0.518220\pi$
$251$	−1.81268	−0.114415	−0.0572077	+	0.998362i	$0.518220\pi$
$252$	0	0
$253$	−7.42010	−0.466498
$254$	0	0
$255$	19.8096	1.24052
$256$	0	0
$257$	1.84493	0.115084	0.0575418	−	0.998343i	$-0.481674\pi$
$257$	1.84493	0.115084	0.0575418	+	0.998343i	$0.481674\pi$
$258$	0	0
$259$	41.3966	2.57226
$260$	0	0
$261$	3.57239	0.221125
$262$	0	0
$263$	7.50485	0.462769	0.231384	−	0.972862i	$-0.425674\pi$
$263$	7.50485	0.462769	0.231384	+	0.972862i	$0.425674\pi$
$264$	0	0
$265$	−2.82879	−0.173771
$266$	0	0
$267$	6.78217	0.415062
$268$	0	0
$269$	−6.31805	−0.385218	−0.192609	−	0.981276i	$-0.561695\pi$
$269$	−6.31805	−0.385218	−0.192609	+	0.981276i	$0.561695\pi$
$270$	0	0
$271$	−8.88653	−0.539818	−0.269909	−	0.962886i	$-0.586994\pi$
$271$	−8.88653	−0.539818	−0.269909	+	0.962886i	$0.586994\pi$
$272$	0	0
$273$	22.8112	1.38060
$274$	0	0
$275$	−12.7955	−0.771596
$276$	0	0
$277$	30.3868	1.82576	0.912882	−	0.408223i	$-0.133851\pi$
$277$	30.3868	1.82576	0.912882	+	0.408223i	$0.133851\pi$
$278$	0	0
$279$	−7.18724	−0.430289
$280$	0	0
$281$	−15.0587	−0.898330	−0.449165	−	0.893449i	$-0.648278\pi$
$281$	−15.0587	−0.898330	−0.449165	+	0.893449i	$0.648278\pi$
$282$	0	0
$283$	−19.8690	−1.18109	−0.590546	−	0.807004i	$-0.701088\pi$
$283$	−19.8690	−1.18109	−0.590546	+	0.807004i	$0.701088\pi$
$284$	0	0
$285$	−2.47354	−0.146520
$286$	0	0
$287$	37.6922	2.22490
$288$	0	0
$289$	13.4388	0.790517
$290$	0	0
$291$	7.41920	0.434921
$292$	0	0
$293$	2.68694	0.156973	0.0784864	−	0.996915i	$-0.474991\pi$
$293$	2.68694	0.156973	0.0784864	+	0.996915i	$0.474991\pi$
$294$	0	0
$295$	9.90472	0.576675
$296$	0	0
$297$	1.62131	0.0940778
$298$	0	0
$299$	25.4117	1.46959
$300$	0	0
$301$	11.8755	0.684493
$302$	0	0
$303$	19.3734	1.11297
$304$	0	0
$305$	−5.74680	−0.329061
$306$	0	0
$307$	−20.8605	−1.19057	−0.595284	−	0.803515i	$-0.702961\pi$
$307$	−20.8605	−1.19057	−0.595284	+	0.803515i	$0.702961\pi$
$308$	0	0
$309$	−2.04851	−0.116536
$310$	0	0
$311$	10.4158	0.590627	0.295313	−	0.955400i	$-0.404576\pi$
$311$	10.4158	0.590627	0.295313	+	0.955400i	$0.404576\pi$
$312$	0	0
$313$	30.9992	1.75218	0.876089	−	0.482149i	$-0.160144\pi$
$313$	30.9992	1.75218	0.876089	+	0.482149i	$0.160144\pi$
$314$	0	0
$315$	−14.7510	−0.831125
$316$	0	0
$317$	−24.8470	−1.39555	−0.697774	−	0.716318i	$-0.745826\pi$
$317$	−24.8470	−1.39555	−0.697774	+	0.716318i	$0.745826\pi$
$318$	0	0
$319$	−5.79194	−0.324286
$320$	0	0
$321$	0.805768	0.0449736
$322$	0	0
$323$	−3.80077	−0.211480
$324$	0	0
$325$	43.8207	2.43074
$326$	0	0
$327$	−8.62091	−0.476737
$328$	0	0
$329$	−23.4305	−1.29176
$330$	0	0
$331$	17.7919	0.977930	0.488965	−	0.872303i	$-0.337375\pi$
$331$	17.7919	0.977930	0.488965	+	0.872303i	$0.337375\pi$
$332$	0	0
$333$	−10.0764	−0.552182
$334$	0	0
$335$	−26.3009	−1.43697
$336$	0	0
$337$	30.5764	1.66560	0.832802	−	0.553570i	$-0.186735\pi$
$337$	30.5764	1.66560	0.832802	+	0.553570i	$0.186735\pi$
$338$	0	0
$339$	−4.93599	−0.268086
$340$	0	0
$341$	11.6527	0.631030
$342$	0	0
$343$	−11.8236	−0.638414
$344$	0	0
$345$	−16.4326	−0.884701
$346$	0	0
$347$	−17.2533	−0.926207	−0.463104	−	0.886304i	$-0.653264\pi$
$347$	−17.2533	−0.926207	−0.463104	+	0.886304i	$0.653264\pi$
$348$	0	0
$349$	−10.6088	−0.567874	−0.283937	−	0.958843i	$-0.591641\pi$
$349$	−10.6088	−0.567874	−0.283937	+	0.958843i	$0.591641\pi$
$350$	0	0
$351$	−5.55250	−0.296370
$352$	0	0
$353$	−25.8287	−1.37472	−0.687362	−	0.726315i	$-0.741231\pi$
$353$	−25.8287	−1.37472	−0.687362	+	0.726315i	$0.741231\pi$
$354$	0	0
$355$	25.0735	1.33076
$356$	0	0
$357$	−22.6660	−1.19961
$358$	0	0
$359$	16.5743	0.874759	0.437379	−	0.899277i	$-0.355907\pi$
$359$	16.5743	0.874759	0.437379	+	0.899277i	$0.355907\pi$
$360$	0	0
$361$	−18.5254	−0.975022
$362$	0	0
$363$	8.37136	0.439383
$364$	0	0
$365$	0.652509	0.0341539
$366$	0	0
$367$	−25.5959	−1.33610	−0.668049	−	0.744117i	$-0.732870\pi$
$367$	−25.5959	−1.33610	−0.668049	+	0.744117i	$0.732870\pi$
$368$	0	0
$369$	−9.17469	−0.477615
$370$	0	0
$371$	3.23668	0.168040
$372$	0	0
$373$	−23.5687	−1.22034	−0.610170	−	0.792270i	$-0.708899\pi$
$373$	−23.5687	−1.22034	−0.610170	+	0.792270i	$0.708899\pi$
$374$	0	0
$375$	−10.3842	−0.536236
$376$	0	0
$377$	19.8357	1.02159
$378$	0	0
$379$	−19.7128	−1.01258	−0.506289	−	0.862364i	$-0.668983\pi$
$379$	−19.7128	−1.01258	−0.506289	+	0.862364i	$0.668983\pi$
$380$	0	0
$381$	−12.3106	−0.630693
$382$	0	0
$383$	10.8187	0.552811	0.276406	−	0.961041i	$-0.410857\pi$
$383$	10.8187	0.552811	0.276406	+	0.961041i	$0.410857\pi$
$384$	0	0
$385$	23.9159	1.21887
$386$	0	0
$387$	−2.89063	−0.146939
$388$	0	0
$389$	36.8399	1.86786	0.933929	−	0.357458i	$-0.116356\pi$
$389$	36.8399	1.86786	0.933929	+	0.357458i	$0.116356\pi$
$390$	0	0
$391$	−25.2498	−1.27694
$392$	0	0
$393$	16.4098	0.827767
$394$	0	0
$395$	49.9883	2.51519
$396$	0	0
$397$	29.3928	1.47518	0.737591	−	0.675247i	$-0.235963\pi$
$397$	29.3928	1.47518	0.737591	+	0.675247i	$0.235963\pi$
$398$	0	0
$399$	2.83021	0.141688
$400$	0	0
$401$	27.4047	1.36853	0.684263	−	0.729235i	$-0.260124\pi$
$401$	27.4047	1.36853	0.684263	+	0.729235i	$0.260124\pi$
$402$	0	0
$403$	−39.9071	−1.98792
$404$	0	0
$405$	3.59055	0.178416
$406$	0	0
$407$	16.3369	0.809790
$408$	0	0
$409$	33.9808	1.68024	0.840120	−	0.542401i	$-0.182484\pi$
$409$	33.9808	1.68024	0.840120	+	0.542401i	$0.182484\pi$
$410$	0	0
$411$	16.2690	0.802492
$412$	0	0
$413$	−11.3329	−0.557655
$414$	0	0
$415$	−13.7231	−0.673640
$416$	0	0
$417$	−5.33754	−0.261381
$418$	0	0
$419$	−15.5038	−0.757408	−0.378704	−	0.925518i	$-0.623630\pi$
$419$	−15.5038	−0.757408	−0.378704	+	0.925518i	$0.623630\pi$
$420$	0	0
$421$	21.4215	1.04402	0.522010	−	0.852939i	$-0.325182\pi$
$421$	21.4215	1.04402	0.522010	+	0.852939i	$0.325182\pi$
$422$	0	0
$423$	5.70322	0.277300
$424$	0	0
$425$	−43.5417	−2.11208
$426$	0	0
$427$	6.57545	0.318208
$428$	0	0
$429$	9.00230	0.434635
$430$	0	0
$431$	−30.5013	−1.46920	−0.734598	−	0.678502i	$-0.762629\pi$
$431$	−30.5013	−1.46920	−0.734598	+	0.678502i	$0.762629\pi$
$432$	0	0
$433$	−27.4355	−1.31847	−0.659234	−	0.751938i	$-0.729119\pi$
$433$	−27.4355	−1.31847	−0.659234	+	0.751938i	$0.729119\pi$
$434$	0	0
$435$	−12.8269	−0.615000
$436$	0	0
$437$	3.15284	0.150821
$438$	0	0
$439$	0.281832	0.0134511	0.00672556	−	0.999977i	$-0.497859\pi$
$439$	0.281832	0.0134511	0.00672556	+	0.999977i	$0.497859\pi$
$440$	0	0
$441$	9.87799	0.470380
$442$	0	0
$443$	11.9068	0.565708	0.282854	−	0.959163i	$-0.408719\pi$
$443$	11.9068	0.565708	0.282854	+	0.959163i	$0.408719\pi$
$444$	0	0
$445$	−24.3517	−1.15438
$446$	0	0
$447$	9.27063	0.438486
$448$	0	0
$449$	18.2052	0.859158	0.429579	−	0.903029i	$-0.358662\pi$
$449$	18.2052	0.859158	0.429579	+	0.903029i	$0.358662\pi$
$450$	0	0
$451$	14.8750	0.700436
$452$	0	0
$453$	−20.4894	−0.962678
$454$	0	0
$455$	−81.9050	−3.83976
$456$	0	0
$457$	−9.31943	−0.435945	−0.217972	−	0.975955i	$-0.569944\pi$
$457$	−9.31943	−0.435945	−0.217972	+	0.975955i	$0.569944\pi$
$458$	0	0
$459$	5.51714	0.257518
$460$	0	0
$461$	−25.0465	−1.16653	−0.583266	−	0.812281i	$-0.698225\pi$
$461$	−25.0465	−1.16653	−0.583266	+	0.812281i	$0.698225\pi$
$462$	0	0
$463$	−0.497940	−0.0231412	−0.0115706	−	0.999933i	$-0.503683\pi$
$463$	−0.497940	−0.0231412	−0.0115706	+	0.999933i	$0.503683\pi$
$464$	0	0
$465$	25.8062	1.19673
$466$	0	0
$467$	−22.1274	−1.02393	−0.511966	−	0.859006i	$-0.671083\pi$
$467$	−22.1274	−1.02393	−0.511966	+	0.859006i	$0.671083\pi$
$468$	0	0
$469$	30.0932	1.38958
$470$	0	0
$471$	3.90511	0.179938
$472$	0	0
$473$	4.68659	0.215490
$474$	0	0
$475$	5.43687	0.249461
$476$	0	0
$477$	−0.787842	−0.0360728
$478$	0	0
$479$	−28.8134	−1.31652	−0.658259	−	0.752792i	$-0.728707\pi$
$479$	−28.8134	−1.31652	−0.658259	+	0.752792i	$0.728707\pi$
$480$	0	0
$481$	−55.9491	−2.55106
$482$	0	0
$483$	18.8020	0.855523
$484$	0	0
$485$	−26.6390	−1.20962
$486$	0	0
$487$	−3.03643	−0.137594	−0.0687968	−	0.997631i	$-0.521916\pi$
$487$	−3.03643	−0.137594	−0.0687968	+	0.997631i	$0.521916\pi$
$488$	0	0
$489$	−13.1543	−0.594856
$490$	0	0
$491$	−20.6390	−0.931425	−0.465713	−	0.884936i	$-0.654202\pi$
$491$	−20.6390	−0.931425	−0.465713	+	0.884936i	$0.654202\pi$
$492$	0	0
$493$	−19.7094	−0.887665
$494$	0	0
$495$	−5.82139	−0.261652
$496$	0	0
$497$	−28.6889	−1.28687
$498$	0	0
$499$	−24.8144	−1.11084	−0.555422	−	0.831568i	$-0.687443\pi$
$499$	−24.8144	−1.11084	−0.555422	+	0.831568i	$0.687443\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	0.0868793	0.00387375	0.00193688	−	0.999998i	$-0.499383\pi$
$503$	0.0868793	0.00387375	0.00193688	+	0.999998i	$0.499383\pi$
$504$	0	0
$505$	−69.5613	−3.09544
$506$	0	0
$507$	−17.8302	−0.791869
$508$	0	0
$509$	35.2337	1.56170	0.780852	−	0.624716i	$-0.214785\pi$
$509$	35.2337	1.56170	0.780852	+	0.624716i	$0.214785\pi$
$510$	0	0
$511$	−0.746596	−0.0330274
$512$	0	0
$513$	−0.688902	−0.0304158
$514$	0	0
$515$	7.35530	0.324113
$516$	0	0
$517$	−9.24667	−0.406668
$518$	0	0
$519$	−11.4082	−0.500766
$520$	0	0
$521$	−19.7270	−0.864257	−0.432129	−	0.901812i	$-0.642237\pi$
$521$	−19.7270	−0.864257	−0.432129	+	0.901812i	$0.642237\pi$
$522$	0	0
$523$	−26.8934	−1.17597	−0.587984	−	0.808872i	$-0.700078\pi$
$523$	−26.8934	−1.17597	−0.587984	+	0.808872i	$0.700078\pi$
$524$	0	0
$525$	32.4229	1.41505
$526$	0	0
$527$	39.6530	1.72731
$528$	0	0
$529$	−2.05457	−0.0893290
$530$	0	0
$531$	2.75855	0.119711
$532$	0	0
$533$	−50.9425	−2.20656
$534$	0	0
$535$	−2.89315	−0.125082
$536$	0	0
$537$	19.7878	0.853905
$538$	0	0
$539$	−16.0153	−0.689826
$540$	0	0
$541$	16.5773	0.712712	0.356356	−	0.934350i	$-0.384019\pi$
$541$	16.5773	0.712712	0.356356	+	0.934350i	$0.384019\pi$
$542$	0	0
$543$	−5.54242	−0.237848
$544$	0	0
$545$	30.9538	1.32592
$546$	0	0
$547$	−23.8486	−1.01969	−0.509846	−	0.860266i	$-0.670298\pi$
$547$	−23.8486	−1.01969	−0.509846	+	0.860266i	$0.670298\pi$
$548$	0	0
$549$	−1.60053	−0.0683091
$550$	0	0
$551$	2.46103	0.104843
$552$	0	0
$553$	−57.1963	−2.43223
$554$	0	0
$555$	36.1798	1.53575
$556$	0	0
$557$	−35.7962	−1.51673	−0.758367	−	0.651828i	$-0.774002\pi$
$557$	−35.7962	−1.51673	−0.758367	+	0.651828i	$0.774002\pi$
$558$	0	0
$559$	−16.0502	−0.678851
$560$	0	0
$561$	−8.94497	−0.377657
$562$	0	0
$563$	18.8666	0.795131	0.397565	−	0.917574i	$-0.369855\pi$
$563$	18.8666	0.795131	0.397565	+	0.917574i	$0.369855\pi$
$564$	0	0
$565$	17.7229	0.745609
$566$	0	0
$567$	−4.10828	−0.172532
$568$	0	0
$569$	−20.0694	−0.841353	−0.420677	−	0.907211i	$-0.638207\pi$
$569$	−20.0694	−0.841353	−0.420677	+	0.907211i	$0.638207\pi$
$570$	0	0
$571$	−37.5976	−1.57341	−0.786706	−	0.617328i	$-0.788215\pi$
$571$	−37.5976	−1.57341	−0.786706	+	0.617328i	$0.788215\pi$
$572$	0	0
$573$	17.3823	0.726158
$574$	0	0
$575$	36.1190	1.50627
$576$	0	0
$577$	−26.4397	−1.10070	−0.550349	−	0.834934i	$-0.685506\pi$
$577$	−26.4397	−1.10070	−0.550349	+	0.834934i	$0.685506\pi$
$578$	0	0
$579$	10.1587	0.422181
$580$	0	0
$581$	15.7018	0.651422
$582$	0	0
$583$	1.27733	0.0529017
$584$	0	0
$585$	19.9365	0.824275
$586$	0	0
$587$	36.6050	1.51085	0.755426	−	0.655234i	$-0.227430\pi$
$587$	36.6050	1.51085	0.755426	+	0.655234i	$0.227430\pi$
$588$	0	0
$589$	−4.95130	−0.204015
$590$	0	0
$591$	−12.6909	−0.522036
$592$	0	0
$593$	−9.38597	−0.385436	−0.192718	−	0.981254i	$-0.561730\pi$
$593$	−9.38597	−0.385436	−0.192718	+	0.981254i	$0.561730\pi$
$594$	0	0
$595$	81.3833	3.33639
$596$	0	0
$597$	27.3399	1.11895
$598$	0	0
$599$	−40.1060	−1.63869	−0.819344	−	0.573303i	$-0.805662\pi$
$599$	−40.1060	−1.63869	−0.819344	+	0.573303i	$0.805662\pi$
$600$	0	0
$601$	31.0025	1.26462	0.632309	−	0.774716i	$-0.282107\pi$
$601$	31.0025	1.26462	0.632309	+	0.774716i	$0.282107\pi$
$602$	0	0
$603$	−7.32502	−0.298298
$604$	0	0
$605$	−30.0578	−1.22202
$606$	0	0
$607$	20.5512	0.834147	0.417074	−	0.908873i	$-0.363056\pi$
$607$	20.5512	0.834147	0.417074	+	0.908873i	$0.363056\pi$
$608$	0	0
$609$	14.6764	0.594717
$610$	0	0
$611$	31.6671	1.28112
$612$	0	0
$613$	36.7012	1.48235	0.741174	−	0.671313i	$-0.234269\pi$
$613$	36.7012	1.48235	0.741174	+	0.671313i	$0.234269\pi$
$614$	0	0
$615$	32.9422	1.32836
$616$	0	0
$617$	−7.21525	−0.290475	−0.145238	−	0.989397i	$-0.546395\pi$
$617$	−7.21525	−0.290475	−0.145238	+	0.989397i	$0.546395\pi$
$618$	0	0
$619$	−1.96526	−0.0789904	−0.0394952	−	0.999220i	$-0.512575\pi$
$619$	−1.96526	−0.0789904	−0.0394952	+	0.999220i	$0.512575\pi$
$620$	0	0
$621$	−4.57662	−0.183653
$622$	0	0
$623$	27.8631	1.11631
$624$	0	0
$625$	−2.17550	−0.0870202
$626$	0	0
$627$	1.11692	0.0446056
$628$	0	0
$629$	55.5927	2.21663
$630$	0	0
$631$	21.5520	0.857970	0.428985	−	0.903312i	$-0.358871\pi$
$631$	21.5520	0.857970	0.428985	+	0.903312i	$0.358871\pi$
$632$	0	0
$633$	17.6413	0.701180
$634$	0	0
$635$	44.2020	1.75410
$636$	0	0
$637$	54.8475	2.17314
$638$	0	0
$639$	6.98319	0.276251
$640$	0	0
$641$	−16.4228	−0.648660	−0.324330	−	0.945944i	$-0.605139\pi$
$641$	−16.4228	−0.648660	−0.324330	+	0.945944i	$0.605139\pi$
$642$	0	0
$643$	−1.24263	−0.0490045	−0.0245023	−	0.999700i	$-0.507800\pi$
$643$	−1.24263	−0.0490045	−0.0245023	+	0.999700i	$0.507800\pi$
$644$	0	0
$645$	10.3790	0.408671
$646$	0	0
$647$	−32.8033	−1.28963	−0.644816	−	0.764338i	$-0.723066\pi$
$647$	−32.8033	−1.28963	−0.644816	+	0.764338i	$0.723066\pi$
$648$	0	0
$649$	−4.47245	−0.175559
$650$	0	0
$651$	−29.5272	−1.15726
$652$	0	0
$653$	10.3674	0.405707	0.202854	−	0.979209i	$-0.434978\pi$
$653$	10.3674	0.405707	0.202854	+	0.979209i	$0.434978\pi$
$654$	0	0
$655$	−58.9204	−2.30221
$656$	0	0
$657$	0.181729	0.00708994
$658$	0	0
$659$	39.4077	1.53511	0.767554	−	0.640984i	$-0.221474\pi$
$659$	39.4077	1.53511	0.767554	+	0.640984i	$0.221474\pi$
$660$	0	0
$661$	−30.3055	−1.17875	−0.589374	−	0.807861i	$-0.700625\pi$
$661$	−30.3055	−1.17875	−0.589374	+	0.807861i	$0.700625\pi$
$662$	0	0
$663$	30.6339	1.18972
$664$	0	0
$665$	−10.1620	−0.394066
$666$	0	0
$667$	16.3495	0.633053
$668$	0	0
$669$	20.6260	0.797445
$670$	0	0
$671$	2.59496	0.100177
$672$	0	0
$673$	−13.8516	−0.533940	−0.266970	−	0.963705i	$-0.586022\pi$
$673$	−13.8516	−0.533940	−0.266970	+	0.963705i	$0.586022\pi$
$674$	0	0
$675$	−7.89208	−0.303766
$676$	0	0
$677$	−25.5715	−0.982791	−0.491396	−	0.870936i	$-0.663513\pi$
$677$	−25.5715	−0.982791	−0.491396	+	0.870936i	$0.663513\pi$
$678$	0	0
$679$	30.4802	1.16972
$680$	0	0
$681$	−6.16425	−0.236214
$682$	0	0
$683$	44.9178	1.71873	0.859365	−	0.511362i	$-0.170859\pi$
$683$	44.9178	1.71873	0.859365	+	0.511362i	$0.170859\pi$
$684$	0	0
$685$	−58.4148	−2.23192
$686$	0	0
$687$	19.4392	0.741652
$688$	0	0
$689$	−4.37449	−0.166655
$690$	0	0
$691$	−6.52688	−0.248294	−0.124147	−	0.992264i	$-0.539619\pi$
$691$	−6.52688	−0.248294	−0.124147	+	0.992264i	$0.539619\pi$
$692$	0	0
$693$	6.66079	0.253022
$694$	0	0
$695$	19.1647	0.726960
$696$	0	0
$697$	50.6180	1.91729
$698$	0	0
$699$	−19.8065	−0.749151
$700$	0	0
$701$	5.51986	0.208482	0.104241	−	0.994552i	$-0.466759\pi$
$701$	5.51986	0.208482	0.104241	+	0.994552i	$0.466759\pi$
$702$	0	0
$703$	−6.94164	−0.261809
$704$	0	0
$705$	−20.4777	−0.771236
$706$	0	0
$707$	79.5914	2.99334
$708$	0	0
$709$	−43.8699	−1.64757	−0.823785	−	0.566902i	$-0.808142\pi$
$709$	−43.8699	−1.64757	−0.823785	+	0.566902i	$0.808142\pi$
$710$	0	0
$711$	13.9222	0.522123
$712$	0	0
$713$	−32.8932	−1.23186
$714$	0	0
$715$	−32.3233	−1.20882
$716$	0	0
$717$	1.11360	0.0415881
$718$	0	0
$719$	16.0080	0.596997	0.298499	−	0.954410i	$-0.403514\pi$
$719$	16.0080	0.596997	0.298499	+	0.954410i	$0.403514\pi$
$720$	0	0
$721$	−8.41587	−0.313423
$722$	0	0
$723$	26.4165	0.982442
$724$	0	0
$725$	28.1936	1.04708
$726$	0	0
$727$	−35.8776	−1.33063	−0.665313	−	0.746565i	$-0.731702\pi$
$727$	−35.8776	−1.33063	−0.665313	+	0.746565i	$0.731702\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.9480	0.589857
$732$	0	0
$733$	−35.6831	−1.31798	−0.658992	−	0.752150i	$-0.729017\pi$
$733$	−35.6831	−1.31798	−0.658992	+	0.752150i	$0.729017\pi$
$734$	0	0
$735$	−35.4675	−1.30824
$736$	0	0
$737$	11.8761	0.437462
$738$	0	0
$739$	36.9496	1.35921	0.679607	−	0.733577i	$-0.262150\pi$
$739$	36.9496	1.35921	0.679607	+	0.733577i	$0.262150\pi$
$740$	0	0
$741$	−3.82513	−0.140520
$742$	0	0
$743$	16.8490	0.618131	0.309066	−	0.951041i	$-0.399984\pi$
$743$	16.8490	0.618131	0.309066	+	0.951041i	$0.399984\pi$
$744$	0	0
$745$	−33.2867	−1.21953
$746$	0	0
$747$	−3.82200	−0.139840
$748$	0	0
$749$	3.31032	0.120957
$750$	0	0
$751$	−45.5008	−1.66035	−0.830174	−	0.557505i	$-0.811759\pi$
$751$	−45.5008	−1.66035	−0.830174	+	0.557505i	$0.811759\pi$
$752$	0	0
$753$	1.81268	0.0660578
$754$	0	0
$755$	73.5684	2.67743
$756$	0	0
$757$	3.61208	0.131283	0.0656417	−	0.997843i	$-0.479091\pi$
$757$	3.61208	0.131283	0.0656417	+	0.997843i	$0.479091\pi$
$758$	0	0
$759$	7.42010	0.269333
$760$	0	0
$761$	31.5545	1.14385	0.571924	−	0.820306i	$-0.306197\pi$
$761$	31.5545	1.14385	0.571924	+	0.820306i	$0.306197\pi$
$762$	0	0
$763$	−35.4171	−1.28219
$764$	0	0
$765$	−19.8096	−0.716217
$766$	0	0
$767$	15.3168	0.553059
$768$	0	0
$769$	14.4083	0.519577	0.259789	−	0.965665i	$-0.416347\pi$
$769$	14.4083	0.519577	0.259789	+	0.965665i	$0.416347\pi$
$770$	0	0
$771$	−1.84493	−0.0664436
$772$	0	0
$773$	6.46104	0.232387	0.116194	−	0.993227i	$-0.462931\pi$
$773$	6.46104	0.232387	0.116194	+	0.993227i	$0.462931\pi$
$774$	0	0
$775$	−56.7222	−2.03752
$776$	0	0
$777$	−41.3966	−1.48510
$778$	0	0
$779$	−6.32047	−0.226454
$780$	0	0
$781$	−11.3219	−0.405129
$782$	0	0
$783$	−3.57239	−0.127667
$784$	0	0
$785$	−14.0215	−0.500449
$786$	0	0
$787$	−15.9998	−0.570330	−0.285165	−	0.958479i	$-0.592048\pi$
$787$	−15.9998	−0.570330	−0.285165	+	0.958479i	$0.592048\pi$
$788$	0	0
$789$	−7.50485	−0.267180
$790$	0	0
$791$	−20.2784	−0.721018
$792$	0	0
$793$	−8.88696	−0.315585
$794$	0	0
$795$	2.82879	0.100327
$796$	0	0
$797$	2.33123	0.0825765	0.0412883	−	0.999147i	$-0.486854\pi$
$797$	2.33123	0.0825765	0.0412883	+	0.999147i	$0.486854\pi$
$798$	0	0
$799$	−31.4655	−1.11317
$800$	0	0
$801$	−6.78217	−0.239636
$802$	0	0
$803$	−0.294639	−0.0103976
$804$	0	0
$805$	−67.5097	−2.37941
$806$	0	0
$807$	6.31805	0.222406
$808$	0	0
$809$	16.0736	0.565117	0.282558	−	0.959250i	$-0.408817\pi$
$809$	16.0736	0.565117	0.282558	+	0.959250i	$0.408817\pi$
$810$	0	0
$811$	−38.6082	−1.35572	−0.677859	−	0.735192i	$-0.737092\pi$
$811$	−38.6082	−1.35572	−0.677859	+	0.735192i	$0.737092\pi$
$812$	0	0
$813$	8.88653	0.311664
$814$	0	0
$815$	47.2311	1.65443
$816$	0	0
$817$	−1.99136	−0.0696689
$818$	0	0
$819$	−22.8112	−0.797089
$820$	0	0
$821$	−8.49502	−0.296478	−0.148239	−	0.988952i	$-0.547361\pi$
$821$	−8.49502	−0.296478	−0.148239	+	0.988952i	$0.547361\pi$
$822$	0	0
$823$	20.2987	0.707567	0.353783	−	0.935327i	$-0.384895\pi$
$823$	20.2987	0.707567	0.353783	+	0.935327i	$0.384895\pi$
$824$	0	0
$825$	12.7955	0.445481
$826$	0	0
$827$	−31.6081	−1.09912	−0.549561	−	0.835454i	$-0.685205\pi$
$827$	−31.6081	−1.09912	−0.549561	+	0.835454i	$0.685205\pi$
$828$	0	0
$829$	30.2482	1.05056	0.525282	−	0.850928i	$-0.323960\pi$
$829$	30.2482	1.05056	0.525282	+	0.850928i	$0.323960\pi$
$830$	0	0
$831$	−30.3868	−1.05411
$832$	0	0
$833$	−54.4982	−1.88825
$834$	0	0
$835$	−3.59055	−0.124256
$836$	0	0
$837$	7.18724	0.248427
$838$	0	0
$839$	38.0441	1.31343	0.656715	−	0.754139i	$-0.271945\pi$
$839$	38.0441	1.31343	0.656715	+	0.754139i	$0.271945\pi$
$840$	0	0
$841$	−16.2380	−0.559933
$842$	0	0
$843$	15.0587	0.518651
$844$	0	0
$845$	64.0204	2.20237
$846$	0	0
$847$	34.3919	1.18172
$848$	0	0
$849$	19.8690	0.681904
$850$	0	0
$851$	−46.1157	−1.58083
$852$	0	0
$853$	14.8978	0.510091	0.255045	−	0.966929i	$-0.417910\pi$
$853$	14.8978	0.510091	0.255045	+	0.966929i	$0.417910\pi$
$854$	0	0
$855$	2.47354	0.0845933
$856$	0	0
$857$	17.2863	0.590487	0.295244	−	0.955422i	$-0.404599\pi$
$857$	17.2863	0.590487	0.295244	+	0.955422i	$0.404599\pi$
$858$	0	0
$859$	−2.45168	−0.0836501	−0.0418250	−	0.999125i	$-0.513317\pi$
$859$	−2.45168	−0.0836501	−0.0418250	+	0.999125i	$0.513317\pi$
$860$	0	0
$861$	−37.6922	−1.28455
$862$	0	0
$863$	−24.4142	−0.831070	−0.415535	−	0.909577i	$-0.636406\pi$
$863$	−24.4142	−0.831070	−0.415535	+	0.909577i	$0.636406\pi$
$864$	0	0
$865$	40.9618	1.39274
$866$	0	0
$867$	−13.4388	−0.456405
$868$	0	0
$869$	−22.5721	−0.765707
$870$	0	0
$871$	−40.6721	−1.37812
$872$	0	0
$873$	−7.41920	−0.251102
$874$	0	0
$875$	−42.6611	−1.44221
$876$	0	0
$877$	−21.2565	−0.717782	−0.358891	−	0.933380i	$-0.616845\pi$
$877$	−21.2565	−0.717782	−0.358891	+	0.933380i	$0.616845\pi$
$878$	0	0
$879$	−2.68694	−0.0906282
$880$	0	0
$881$	−22.9501	−0.773209	−0.386605	−	0.922246i	$-0.626352\pi$
$881$	−22.9501	−0.773209	−0.386605	+	0.922246i	$0.626352\pi$
$882$	0	0
$883$	−5.97231	−0.200984	−0.100492	−	0.994938i	$-0.532042\pi$
$883$	−5.97231	−0.200984	−0.100492	+	0.994938i	$0.532042\pi$
$884$	0	0
$885$	−9.90472	−0.332943
$886$	0	0
$887$	6.82725	0.229237	0.114618	−	0.993410i	$-0.463436\pi$
$887$	6.82725	0.229237	0.114618	+	0.993410i	$0.463436\pi$
$888$	0	0
$889$	−50.5755	−1.69625
$890$	0	0
$891$	−1.62131	−0.0543158
$892$	0	0
$893$	3.92896	0.131478
$894$	0	0
$895$	−71.0490	−2.37491
$896$	0	0
$897$	−25.4117	−0.848471
$898$	0	0
$899$	−25.6756	−0.856329
$900$	0	0
$901$	4.34663	0.144807
$902$	0	0
$903$	−11.8755	−0.395192
$904$	0	0
$905$	19.9004	0.661510
$906$	0	0
$907$	−16.0600	−0.533263	−0.266632	−	0.963798i	$-0.585911\pi$
$907$	−16.0600	−0.533263	−0.266632	+	0.963798i	$0.585911\pi$
$908$	0	0
$909$	−19.3734	−0.642575
$910$	0	0
$911$	−31.0437	−1.02852	−0.514262	−	0.857633i	$-0.671934\pi$
$911$	−31.0437	−1.02852	−0.514262	+	0.857633i	$0.671934\pi$
$912$	0	0
$913$	6.19663	0.205079
$914$	0	0
$915$	5.74680	0.189983
$916$	0	0
$917$	67.4162	2.22628
$918$	0	0
$919$	37.2155	1.22762	0.613812	−	0.789452i	$-0.289635\pi$
$919$	37.2155	1.22762	0.613812	+	0.789452i	$0.289635\pi$
$920$	0	0
$921$	20.8605	0.687375
$922$	0	0
$923$	38.7741	1.27627
$924$	0	0
$925$	−79.5235	−2.61472
$926$	0	0
$927$	2.04851	0.0672820
$928$	0	0
$929$	−7.11946	−0.233582	−0.116791	−	0.993157i	$-0.537261\pi$
$929$	−7.11946	−0.233582	−0.116791	+	0.993157i	$0.537261\pi$
$930$	0	0
$931$	6.80497	0.223024
$932$	0	0
$933$	−10.4158	−0.340999
$934$	0	0
$935$	32.1174	1.05035
$936$	0	0
$937$	−14.9767	−0.489268	−0.244634	−	0.969616i	$-0.578668\pi$
$937$	−14.9767	−0.489268	−0.244634	+	0.969616i	$0.578668\pi$
$938$	0	0
$939$	−30.9992	−1.01162
$940$	0	0
$941$	−48.7595	−1.58951	−0.794756	−	0.606929i	$-0.792401\pi$
$941$	−48.7595	−1.58951	−0.794756	+	0.606929i	$0.792401\pi$
$942$	0	0
$943$	−41.9891	−1.36735
$944$	0	0
$945$	14.7510	0.479850
$946$	0	0
$947$	18.6838	0.607142	0.303571	−	0.952809i	$-0.401821\pi$
$947$	18.6838	0.607142	0.303571	+	0.952809i	$0.401821\pi$
$948$	0	0
$949$	1.00905	0.0327552
$950$	0	0
$951$	24.8470	0.805720
$952$	0	0
$953$	57.8655	1.87445	0.937224	−	0.348729i	$-0.113387\pi$
$953$	57.8655	1.87445	0.937224	+	0.348729i	$0.113387\pi$
$954$	0	0
$955$	−62.4122	−2.01961
$956$	0	0
$957$	5.79194	0.187227
$958$	0	0
$959$	66.8378	2.15831
$960$	0	0
$961$	20.6564	0.666335
$962$	0	0
$963$	−0.805768	−0.0259655
$964$	0	0
$965$	−36.4753	−1.17418
$966$	0	0
$967$	13.9913	0.449930	0.224965	−	0.974367i	$-0.427773\pi$
$967$	13.9913	0.449930	0.224965	+	0.974367i	$0.427773\pi$
$968$	0	0
$969$	3.80077	0.122098
$970$	0	0
$971$	45.1729	1.44967	0.724833	−	0.688924i	$-0.241917\pi$
$971$	45.1729	1.44967	0.724833	+	0.688924i	$0.241917\pi$
$972$	0	0
$973$	−21.9281	−0.702984
$974$	0	0
$975$	−43.8207	−1.40339
$976$	0	0
$977$	4.49820	0.143910	0.0719550	−	0.997408i	$-0.477076\pi$
$977$	4.49820	0.143910	0.0719550	+	0.997408i	$0.477076\pi$
$978$	0	0
$979$	10.9960	0.351433
$980$	0	0
$981$	8.62091	0.275244
$982$	0	0
$983$	−6.41141	−0.204492	−0.102246	−	0.994759i	$-0.532603\pi$
$983$	−6.41141	−0.204492	−0.102246	+	0.994759i	$0.532603\pi$
$984$	0	0
$985$	45.5675	1.45190
$986$	0	0
$987$	23.4305	0.745800
$988$	0	0
$989$	−13.2293	−0.420667
$990$	0	0
$991$	42.9219	1.36346	0.681730	−	0.731604i	$-0.261228\pi$
$991$	42.9219	1.36346	0.681730	+	0.731604i	$0.261228\pi$
$992$	0	0
$993$	−17.7919	−0.564608
$994$	0	0
$995$	−98.1655	−3.11205
$996$	0	0
$997$	−0.859857	−0.0272319	−0.0136160	−	0.999907i	$-0.504334\pi$
$997$	−0.859857	−0.0272319	−0.0136160	+	0.999907i	$0.504334\pi$
$998$	0	0
$999$	10.0764	0.318802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bf.1.11		12
4.3	odd	2		4008.2.a.l.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.11	✓	12	4.3	odd	2
8016.2.a.bf.1.11		12	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.59055 q^{5} -4.10828 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.59055 q^{5} -4.10828 q^{7} +1.00000 q^{9} -1.62131 q^{11} +5.55250 q^{13} -3.59055 q^{15} -5.51714 q^{17} +0.688902 q^{19} +4.10828 q^{21} +4.57662 q^{23} +7.89208 q^{25} -1.00000 q^{27} +3.57239 q^{29} -7.18724 q^{31} +1.62131 q^{33} -14.7510 q^{35} -10.0764 q^{37} -5.55250 q^{39} -9.17469 q^{41} -2.89063 q^{43} +3.59055 q^{45} +5.70322 q^{47} +9.87799 q^{49} +5.51714 q^{51} -0.787842 q^{53} -5.82139 q^{55} -0.688902 q^{57} +2.75855 q^{59} -1.60053 q^{61} -4.10828 q^{63} +19.9365 q^{65} -7.32502 q^{67} -4.57662 q^{69} +6.98319 q^{71} +0.181729 q^{73} -7.89208 q^{75} +6.66079 q^{77} +13.9222 q^{79} +1.00000 q^{81} -3.82200 q^{83} -19.8096 q^{85} -3.57239 q^{87} -6.78217 q^{89} -22.8112 q^{91} +7.18724 q^{93} +2.47354 q^{95} -7.41920 q^{97} -1.62131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9 - q^11 + 8 * q^13 - 4 * q^15 + 3 * q^17 - 12 * q^19 + 11 * q^21 - 7 * q^23 + 18 * q^25 - 12 * q^27 + 5 * q^29 - 33 * q^31 + q^33 - 15 * q^35 + 8 * q^37 - 8 * q^39 - 6 * q^41 - 16 * q^43 + 4 * q^45 - 18 * q^47 + 25 * q^49 - 3 * q^51 + 20 * q^53 - 39 * q^55 + 12 * q^57 - 4 * q^59 + 10 * q^61 - 11 * q^63 - 9 * q^67 + 7 * q^69 - 11 * q^71 + 22 * q^73 - 18 * q^75 + 24 * q^77 - 56 * q^79 + 12 * q^81 - 26 * q^83 + 15 * q^85 - 5 * q^87 - 15 * q^89 - 11 * q^91 + 33 * q^93 - 3 * q^95 + 8 * q^97 - q^99

Introduction

L-functions

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Groups

Database

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