LMFDB - Embedded newform 8016.2.a.bb.1.2

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:	$0$
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} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 $ x^9 - 29x^7 - 7x^6 + 266x^5 + 69x^4 - 901x^3 - 199x^2 + 875*x + 391
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.69402$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.69402 q^{5} +4.12928 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.69402 q^{5} +4.12928 q^{7} +1.00000 q^{9} -2.48865 q^{11} +1.35347 q^{13} +1.69402 q^{15} -7.89205 q^{17} -5.08686 q^{19} -4.12928 q^{21} -6.55829 q^{23} -2.13028 q^{25} -1.00000 q^{27} +7.41203 q^{29} -4.69564 q^{31} +2.48865 q^{33} -6.99510 q^{35} -0.0133728 q^{37} -1.35347 q^{39} -1.02270 q^{41} +8.67968 q^{43} -1.69402 q^{45} -5.31896 q^{47} +10.0509 q^{49} +7.89205 q^{51} +12.1374 q^{53} +4.21584 q^{55} +5.08686 q^{57} +0.0660655 q^{59} +10.3680 q^{61} +4.12928 q^{63} -2.29282 q^{65} -6.10634 q^{67} +6.55829 q^{69} +9.95916 q^{71} -3.76173 q^{73} +2.13028 q^{75} -10.2764 q^{77} -11.4241 q^{79} +1.00000 q^{81} -10.4077 q^{83} +13.3693 q^{85} -7.41203 q^{87} +6.46475 q^{89} +5.58886 q^{91} +4.69564 q^{93} +8.61727 q^{95} +18.7791 q^{97} -2.48865 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9} - 7 q^{11} + 6 q^{13} - 9 q^{15} + 7 q^{17} - 2 q^{19} + 2 q^{21} - 19 q^{23} + 22 q^{25} - 9 q^{27} + 13 q^{29} - 12 q^{31} + 7 q^{33} - 4 q^{35} + 15 q^{37} - 6 q^{39} + 18 q^{41} + 6 q^{43} + 9 q^{45} - 25 q^{47} + 19 q^{49} - 7 q^{51} + 17 q^{53} + 3 q^{55} + 2 q^{57} - 3 q^{59} + 14 q^{61} - 2 q^{63} + 14 q^{65} + 4 q^{67} + 19 q^{69} - 17 q^{71} - 20 q^{73} - 22 q^{75} + 14 q^{77} + 8 q^{79} + 9 q^{81} + q^{83} + 5 q^{85} - 13 q^{87} + 36 q^{89} + 41 q^{91} + 12 q^{93} - 5 q^{95} + 31 q^{97} - 7 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9 - 7 * q^11 + 6 * q^13 - 9 * q^15 + 7 * q^17 - 2 * q^19 + 2 * q^21 - 19 * q^23 + 22 * q^25 - 9 * q^27 + 13 * q^29 - 12 * q^31 + 7 * q^33 - 4 * q^35 + 15 * q^37 - 6 * q^39 + 18 * q^41 + 6 * q^43 + 9 * q^45 - 25 * q^47 + 19 * q^49 - 7 * q^51 + 17 * q^53 + 3 * q^55 + 2 * q^57 - 3 * q^59 + 14 * q^61 - 2 * q^63 + 14 * q^65 + 4 * q^67 + 19 * q^69 - 17 * q^71 - 20 * q^73 - 22 * q^75 + 14 * q^77 + 8 * q^79 + 9 * q^81 + q^83 + 5 * q^85 - 13 * q^87 + 36 * q^89 + 41 * q^91 + 12 * q^93 - 5 * q^95 + 31 * q^97 - 7 * 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
$5$	−1.69402	−0.757591	−0.378795	−	0.925480i	$-0.623662\pi$
$5$	−1.69402	−0.757591	−0.378795	+	0.925480i	$0.623662\pi$
$6$	0	0
$7$	4.12928	1.56072	0.780360	−	0.625330i	$-0.215036\pi$
$7$	4.12928	1.56072	0.780360	+	0.625330i	$0.215036\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.48865	−0.750358	−0.375179	−	0.926952i	$-0.622419\pi$
$11$	−2.48865	−0.750358	−0.375179	+	0.926952i	$0.622419\pi$
$12$	0	0
$13$	1.35347	0.375386	0.187693	−	0.982228i	$-0.439899\pi$
$13$	1.35347	0.375386	0.187693	+	0.982228i	$0.439899\pi$
$14$	0	0
$15$	1.69402	0.437395
$16$	0	0
$17$	−7.89205	−1.91410	−0.957052	−	0.289918i	$-0.906372\pi$
$17$	−7.89205	−1.91410	−0.957052	+	0.289918i	$0.906372\pi$
$18$	0	0
$19$	−5.08686	−1.16701	−0.583503	−	0.812111i	$-0.698318\pi$
$19$	−5.08686	−1.16701	−0.583503	+	0.812111i	$0.698318\pi$
$20$	0	0
$21$	−4.12928	−0.901083
$22$	0	0
$23$	−6.55829	−1.36750	−0.683749	−	0.729717i	$-0.739652\pi$
$23$	−6.55829	−1.36750	−0.683749	+	0.729717i	$0.739652\pi$
$24$	0	0
$25$	−2.13028	−0.426056
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.41203	1.37638	0.688189	−	0.725531i	$-0.258406\pi$
$29$	7.41203	1.37638	0.688189	+	0.725531i	$0.258406\pi$
$30$	0	0
$31$	−4.69564	−0.843361	−0.421681	−	0.906744i	$-0.638560\pi$
$31$	−4.69564	−0.843361	−0.421681	+	0.906744i	$0.638560\pi$
$32$	0	0
$33$	2.48865	0.433219
$34$	0	0
$35$	−6.99510	−1.18239
$36$	0	0
$37$	−0.0133728	−0.00219847	−0.00109924	−	0.999999i	$-0.500350\pi$
$37$	−0.0133728	−0.00219847	−0.00109924	+	0.999999i	$0.500350\pi$
$38$	0	0
$39$	−1.35347	−0.216729
$40$	0	0
$41$	−1.02270	−0.159719	−0.0798593	−	0.996806i	$-0.525447\pi$
$41$	−1.02270	−0.159719	−0.0798593	+	0.996806i	$0.525447\pi$
$42$	0	0
$43$	8.67968	1.32364	0.661819	−	0.749663i	$-0.269785\pi$
$43$	8.67968	1.32364	0.661819	+	0.749663i	$0.269785\pi$
$44$	0	0
$45$	−1.69402	−0.252530
$46$	0	0
$47$	−5.31896	−0.775850	−0.387925	−	0.921691i	$-0.626808\pi$
$47$	−5.31896	−0.775850	−0.387925	+	0.921691i	$0.626808\pi$
$48$	0	0
$49$	10.0509	1.43585
$50$	0	0
$51$	7.89205	1.10511
$52$	0	0
$53$	12.1374	1.66720	0.833601	−	0.552367i	$-0.186275\pi$
$53$	12.1374	1.66720	0.833601	+	0.552367i	$0.186275\pi$
$54$	0	0
$55$	4.21584	0.568464
$56$	0	0
$57$	5.08686	0.673772
$58$	0	0
$59$	0.0660655	0.00860099	0.00430050	−	0.999991i	$-0.498631\pi$
$59$	0.0660655	0.00860099	0.00430050	+	0.999991i	$0.498631\pi$
$60$	0	0
$61$	10.3680	1.32749	0.663746	−	0.747958i	$-0.268966\pi$
$61$	10.3680	1.32749	0.663746	+	0.747958i	$0.268966\pi$
$62$	0	0
$63$	4.12928	0.520240
$64$	0	0
$65$	−2.29282	−0.284389
$66$	0	0
$67$	−6.10634	−0.746008	−0.373004	−	0.927830i	$-0.621672\pi$
$67$	−6.10634	−0.746008	−0.373004	+	0.927830i	$0.621672\pi$
$68$	0	0
$69$	6.55829	0.789526
$70$	0	0
$71$	9.95916	1.18193	0.590967	−	0.806696i	$-0.298746\pi$
$71$	9.95916	1.18193	0.590967	+	0.806696i	$0.298746\pi$
$72$	0	0
$73$	−3.76173	−0.440277	−0.220139	−	0.975469i	$-0.570651\pi$
$73$	−3.76173	−0.440277	−0.220139	+	0.975469i	$0.570651\pi$
$74$	0	0
$75$	2.13028	0.245984
$76$	0	0
$77$	−10.2764	−1.17110
$78$	0	0
$79$	−11.4241	−1.28531	−0.642655	−	0.766155i	$-0.722167\pi$
$79$	−11.4241	−1.28531	−0.642655	+	0.766155i	$0.722167\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.4077	−1.14240	−0.571199	−	0.820811i	$-0.693522\pi$
$83$	−10.4077	−1.14240	−0.571199	+	0.820811i	$0.693522\pi$
$84$	0	0
$85$	13.3693	1.45011
$86$	0	0
$87$	−7.41203	−0.794653
$88$	0	0
$89$	6.46475	0.685262	0.342631	−	0.939470i	$-0.388682\pi$
$89$	6.46475	0.685262	0.342631	+	0.939470i	$0.388682\pi$
$90$	0	0
$91$	5.58886	0.585872
$92$	0	0
$93$	4.69564	0.486915
$94$	0	0
$95$	8.61727	0.884114
$96$	0	0
$97$	18.7791	1.90673	0.953364	−	0.301822i	$-0.0975947\pi$
$97$	18.7791	1.90673	0.953364	+	0.301822i	$0.0975947\pi$
$98$	0	0
$99$	−2.48865	−0.250119
$100$	0	0
$101$	14.0599	1.39901	0.699507	−	0.714625i	$-0.253403\pi$
$101$	14.0599	1.39901	0.699507	+	0.714625i	$0.253403\pi$
$102$	0	0
$103$	17.8777	1.76154	0.880770	−	0.473545i	$-0.157026\pi$
$103$	17.8777	1.76154	0.880770	+	0.473545i	$0.157026\pi$
$104$	0	0
$105$	6.99510	0.682652
$106$	0	0
$107$	−13.1294	−1.26926	−0.634632	−	0.772815i	$-0.718848\pi$
$107$	−13.1294	−1.26926	−0.634632	+	0.772815i	$0.718848\pi$
$108$	0	0
$109$	8.71046	0.834311	0.417155	−	0.908835i	$-0.363027\pi$
$109$	8.71046	0.834311	0.417155	+	0.908835i	$0.363027\pi$
$110$	0	0
$111$	0.0133728	0.00126929
$112$	0	0
$113$	−3.25013	−0.305747	−0.152873	−	0.988246i	$-0.548853\pi$
$113$	−3.25013	−0.305747	−0.152873	+	0.988246i	$0.548853\pi$
$114$	0	0
$115$	11.1099	1.03600
$116$	0	0
$117$	1.35347	0.125129
$118$	0	0
$119$	−32.5885	−2.98738
$120$	0	0
$121$	−4.80660	−0.436963
$122$	0	0
$123$	1.02270	0.0922136
$124$	0	0
$125$	12.0789	1.08037
$126$	0	0
$127$	16.6967	1.48159	0.740797	−	0.671729i	$-0.234448\pi$
$127$	16.6967	1.48159	0.740797	+	0.671729i	$0.234448\pi$
$128$	0	0
$129$	−8.67968	−0.764203
$130$	0	0
$131$	9.32465	0.814699	0.407349	−	0.913272i	$-0.366453\pi$
$131$	9.32465	0.814699	0.407349	+	0.913272i	$0.366453\pi$
$132$	0	0
$133$	−21.0051	−1.82137
$134$	0	0
$135$	1.69402	0.145798
$136$	0	0
$137$	−22.0786	−1.88630	−0.943150	−	0.332367i	$-0.892153\pi$
$137$	−22.0786	−1.88630	−0.943150	+	0.332367i	$0.892153\pi$
$138$	0	0
$139$	−1.95579	−0.165888	−0.0829441	−	0.996554i	$-0.526432\pi$
$139$	−1.95579	−0.165888	−0.0829441	+	0.996554i	$0.526432\pi$
$140$	0	0
$141$	5.31896	0.447937
$142$	0	0
$143$	−3.36832	−0.281673
$144$	0	0
$145$	−12.5562	−1.04273
$146$	0	0
$147$	−10.0509	−0.828988
$148$	0	0
$149$	−1.90601	−0.156146	−0.0780730	−	0.996948i	$-0.524877\pi$
$149$	−1.90601	−0.156146	−0.0780730	+	0.996948i	$0.524877\pi$
$150$	0	0
$151$	6.47280	0.526749	0.263374	−	0.964694i	$-0.415165\pi$
$151$	6.47280	0.526749	0.263374	+	0.964694i	$0.415165\pi$
$152$	0	0
$153$	−7.89205	−0.638034
$154$	0	0
$155$	7.95452	0.638923
$156$	0	0
$157$	0.438526	0.0349982	0.0174991	−	0.999847i	$-0.494430\pi$
$157$	0.438526	0.0349982	0.0174991	+	0.999847i	$0.494430\pi$
$158$	0	0
$159$	−12.1374	−0.962560
$160$	0	0
$161$	−27.0810	−2.13428
$162$	0	0
$163$	12.2469	0.959252	0.479626	−	0.877473i	$-0.340772\pi$
$163$	12.2469	0.959252	0.479626	+	0.877473i	$0.340772\pi$
$164$	0	0
$165$	−4.21584	−0.328203
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−11.1681	−0.859086
$170$	0	0
$171$	−5.08686	−0.389002
$172$	0	0
$173$	8.71361	0.662483	0.331242	−	0.943546i	$-0.392532\pi$
$173$	8.71361	0.662483	0.331242	+	0.943546i	$0.392532\pi$
$174$	0	0
$175$	−8.79652	−0.664955
$176$	0	0
$177$	−0.0660655	−0.00496579
$178$	0	0
$179$	−21.3884	−1.59864	−0.799322	−	0.600903i	$-0.794808\pi$
$179$	−21.3884	−1.59864	−0.799322	+	0.600903i	$0.794808\pi$
$180$	0	0
$181$	−1.85644	−0.137988	−0.0689942	−	0.997617i	$-0.521979\pi$
$181$	−1.85644	−0.137988	−0.0689942	+	0.997617i	$0.521979\pi$
$182$	0	0
$183$	−10.3680	−0.766428
$184$	0	0
$185$	0.0226538	0.00166554
$186$	0	0
$187$	19.6406	1.43626
$188$	0	0
$189$	−4.12928	−0.300361
$190$	0	0
$191$	−11.4811	−0.830746	−0.415373	−	0.909651i	$-0.636349\pi$
$191$	−11.4811	−0.830746	−0.415373	+	0.909651i	$0.636349\pi$
$192$	0	0
$193$	−25.0096	−1.80023	−0.900115	−	0.435653i	$-0.856518\pi$
$193$	−25.0096	−1.80023	−0.900115	+	0.435653i	$0.856518\pi$
$194$	0	0
$195$	2.29282	0.164192
$196$	0	0
$197$	25.2259	1.79727	0.898634	−	0.438699i	$-0.144561\pi$
$197$	25.2259	1.79727	0.898634	+	0.438699i	$0.144561\pi$
$198$	0	0
$199$	25.2035	1.78663	0.893315	−	0.449431i	$-0.148373\pi$
$199$	25.2035	1.78663	0.893315	+	0.449431i	$0.148373\pi$
$200$	0	0
$201$	6.10634	0.430708
$202$	0	0
$203$	30.6063	2.14814
$204$	0	0
$205$	1.73248	0.121001
$206$	0	0
$207$	−6.55829	−0.455833
$208$	0	0
$209$	12.6594	0.875672
$210$	0	0
$211$	5.16157	0.355337	0.177669	−	0.984090i	$-0.443145\pi$
$211$	5.16157	0.355337	0.177669	+	0.984090i	$0.443145\pi$
$212$	0	0
$213$	−9.95916	−0.682390
$214$	0	0
$215$	−14.7036	−1.00278
$216$	0	0
$217$	−19.3896	−1.31625
$218$	0	0
$219$	3.76173	0.254194
$220$	0	0
$221$	−10.6817	−0.718527
$222$	0	0
$223$	−2.36856	−0.158610	−0.0793052	−	0.996850i	$-0.525270\pi$
$223$	−2.36856	−0.158610	−0.0793052	+	0.996850i	$0.525270\pi$
$224$	0	0
$225$	−2.13028	−0.142019
$226$	0	0
$227$	0.662698	0.0439848	0.0219924	−	0.999758i	$-0.492999\pi$
$227$	0.662698	0.0439848	0.0219924	+	0.999758i	$0.492999\pi$
$228$	0	0
$229$	28.4020	1.87686	0.938428	−	0.345475i	$-0.112282\pi$
$229$	28.4020	1.87686	0.938428	+	0.345475i	$0.112282\pi$
$230$	0	0
$231$	10.2764	0.676134
$232$	0	0
$233$	1.11158	0.0728218	0.0364109	−	0.999337i	$-0.488407\pi$
$233$	1.11158	0.0728218	0.0364109	+	0.999337i	$0.488407\pi$
$234$	0	0
$235$	9.01045	0.587777
$236$	0	0
$237$	11.4241	0.742074
$238$	0	0
$239$	5.03519	0.325700	0.162850	−	0.986651i	$-0.447931\pi$
$239$	5.03519	0.325700	0.162850	+	0.986651i	$0.447931\pi$
$240$	0	0
$241$	15.9384	1.02668	0.513341	−	0.858185i	$-0.328408\pi$
$241$	15.9384	1.02668	0.513341	+	0.858185i	$0.328408\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−17.0266	−1.08779
$246$	0	0
$247$	−6.88493	−0.438078
$248$	0	0
$249$	10.4077	0.659564
$250$	0	0
$251$	−12.4875	−0.788205	−0.394102	−	0.919067i	$-0.628944\pi$
$251$	−12.4875	−0.788205	−0.394102	+	0.919067i	$0.628944\pi$
$252$	0	0
$253$	16.3213	1.02611
$254$	0	0
$255$	−13.3693	−0.837220
$256$	0	0
$257$	13.4024	0.836016	0.418008	−	0.908443i	$-0.362728\pi$
$257$	13.4024	0.836016	0.418008	+	0.908443i	$0.362728\pi$
$258$	0	0
$259$	−0.0552200	−0.00343120
$260$	0	0
$261$	7.41203	0.458793
$262$	0	0
$263$	−10.2710	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$263$	−10.2710	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$264$	0	0
$265$	−20.5611	−1.26306
$266$	0	0
$267$	−6.46475	−0.395636
$268$	0	0
$269$	21.0795	1.28524	0.642619	−	0.766186i	$-0.277848\pi$
$269$	21.0795	1.28524	0.642619	+	0.766186i	$0.277848\pi$
$270$	0	0
$271$	22.7180	1.38002	0.690011	−	0.723799i	$-0.257606\pi$
$271$	22.7180	1.38002	0.690011	+	0.723799i	$0.257606\pi$
$272$	0	0
$273$	−5.58886	−0.338253
$274$	0	0
$275$	5.30153	0.319694
$276$	0	0
$277$	−25.7334	−1.54617	−0.773084	−	0.634303i	$-0.781287\pi$
$277$	−25.7334	−1.54617	−0.773084	+	0.634303i	$0.781287\pi$
$278$	0	0
$279$	−4.69564	−0.281120
$280$	0	0
$281$	−20.0242	−1.19454	−0.597271	−	0.802039i	$-0.703748\pi$
$281$	−20.0242	−1.19454	−0.597271	+	0.802039i	$0.703748\pi$
$282$	0	0
$283$	7.01267	0.416860	0.208430	−	0.978037i	$-0.433165\pi$
$283$	7.01267	0.416860	0.208430	+	0.978037i	$0.433165\pi$
$284$	0	0
$285$	−8.61727	−0.510443
$286$	0	0
$287$	−4.22301	−0.249276
$288$	0	0
$289$	45.2844	2.66379
$290$	0	0
$291$	−18.7791	−1.10085
$292$	0	0
$293$	2.63359	0.153856	0.0769279	−	0.997037i	$-0.475489\pi$
$293$	2.63359	0.153856	0.0769279	+	0.997037i	$0.475489\pi$
$294$	0	0
$295$	−0.111917	−0.00651603
$296$	0	0
$297$	2.48865	0.144406
$298$	0	0
$299$	−8.87647	−0.513339
$300$	0	0
$301$	35.8408	2.06583
$302$	0	0
$303$	−14.0599	−0.807722
$304$	0	0
$305$	−17.5637	−1.00570
$306$	0	0
$307$	−12.3105	−0.702597	−0.351299	−	0.936263i	$-0.614260\pi$
$307$	−12.3105	−0.702597	−0.351299	+	0.936263i	$0.614260\pi$
$308$	0	0
$309$	−17.8777	−1.01703
$310$	0	0
$311$	−23.9999	−1.36091	−0.680453	−	0.732791i	$-0.738217\pi$
$311$	−23.9999	−1.36091	−0.680453	+	0.732791i	$0.738217\pi$
$312$	0	0
$313$	3.38113	0.191113	0.0955564	−	0.995424i	$-0.469537\pi$
$313$	3.38113	0.191113	0.0955564	+	0.995424i	$0.469537\pi$
$314$	0	0
$315$	−6.99510	−0.394129
$316$	0	0
$317$	−7.85532	−0.441199	−0.220599	−	0.975365i	$-0.570801\pi$
$317$	−7.85532	−0.441199	−0.220599	+	0.975365i	$0.570801\pi$
$318$	0	0
$319$	−18.4460	−1.03278
$320$	0	0
$321$	13.1294	0.732810
$322$	0	0
$323$	40.1458	2.23377
$324$	0	0
$325$	−2.88327	−0.159935
$326$	0	0
$327$	−8.71046	−0.481689
$328$	0	0
$329$	−21.9635	−1.21089
$330$	0	0
$331$	24.1080	1.32510	0.662548	−	0.749019i	$-0.269475\pi$
$331$	24.1080	1.32510	0.662548	+	0.749019i	$0.269475\pi$
$332$	0	0
$333$	−0.0133728	−0.000732825 0
$334$	0	0
$335$	10.3443	0.565169
$336$	0	0
$337$	12.7093	0.692321	0.346161	−	0.938175i	$-0.387485\pi$
$337$	12.7093	0.692321	0.346161	+	0.938175i	$0.387485\pi$
$338$	0	0
$339$	3.25013	0.176523
$340$	0	0
$341$	11.6858	0.632822
$342$	0	0
$343$	12.5982	0.680240
$344$	0	0
$345$	−11.1099	−0.598138
$346$	0	0
$347$	21.7727	1.16882	0.584410	−	0.811459i	$-0.301326\pi$
$347$	21.7727	1.16882	0.584410	+	0.811459i	$0.301326\pi$
$348$	0	0
$349$	−35.7390	−1.91307	−0.956533	−	0.291625i	$-0.905804\pi$
$349$	−35.7390	−1.91307	−0.956533	+	0.291625i	$0.905804\pi$
$350$	0	0
$351$	−1.35347	−0.0722430
$352$	0	0
$353$	−3.84039	−0.204403	−0.102202	−	0.994764i	$-0.532589\pi$
$353$	−3.84039	−0.204403	−0.102202	+	0.994764i	$0.532589\pi$
$354$	0	0
$355$	−16.8711	−0.895423
$356$	0	0
$357$	32.5885	1.72477
$358$	0	0
$359$	19.9436	1.05258	0.526292	−	0.850304i	$-0.323582\pi$
$359$	19.9436	1.05258	0.526292	+	0.850304i	$0.323582\pi$
$360$	0	0
$361$	6.87619	0.361905
$362$	0	0
$363$	4.80660	0.252281
$364$	0	0
$365$	6.37246	0.333550
$366$	0	0
$367$	11.6497	0.608108	0.304054	−	0.952655i	$-0.401660\pi$
$367$	11.6497	0.608108	0.304054	+	0.952655i	$0.401660\pi$
$368$	0	0
$369$	−1.02270	−0.0532395
$370$	0	0
$371$	50.1188	2.60204
$372$	0	0
$373$	−14.0130	−0.725568	−0.362784	−	0.931873i	$-0.618174\pi$
$373$	−14.0130	−0.725568	−0.362784	+	0.931873i	$0.618174\pi$
$374$	0	0
$375$	−12.0789	−0.623750
$376$	0	0
$377$	10.0320	0.516673
$378$	0	0
$379$	−3.20637	−0.164700	−0.0823502	−	0.996603i	$-0.526243\pi$
$379$	−3.20637	−0.164700	−0.0823502	+	0.996603i	$0.526243\pi$
$380$	0	0
$381$	−16.6967	−0.855398
$382$	0	0
$383$	1.38840	0.0709437	0.0354719	−	0.999371i	$-0.488707\pi$
$383$	1.38840	0.0709437	0.0354719	+	0.999371i	$0.488707\pi$
$384$	0	0
$385$	17.4084	0.887214
$386$	0	0
$387$	8.67968	0.441213
$388$	0	0
$389$	−18.6756	−0.946891	−0.473446	−	0.880823i	$-0.656990\pi$
$389$	−18.6756	−0.946891	−0.473446	+	0.880823i	$0.656990\pi$
$390$	0	0
$391$	51.7584	2.61753
$392$	0	0
$393$	−9.32465	−0.470367
$394$	0	0
$395$	19.3527	0.973740
$396$	0	0
$397$	37.4874	1.88144	0.940719	−	0.339187i	$-0.110152\pi$
$397$	37.4874	1.88144	0.940719	+	0.339187i	$0.110152\pi$
$398$	0	0
$399$	21.0051	1.05157
$400$	0	0
$401$	17.6036	0.879084	0.439542	−	0.898222i	$-0.355141\pi$
$401$	17.6036	0.879084	0.439542	+	0.898222i	$0.355141\pi$
$402$	0	0
$403$	−6.35541	−0.316586
$404$	0	0
$405$	−1.69402	−0.0841768
$406$	0	0
$407$	0.0332803	0.00164964
$408$	0	0
$409$	6.15357	0.304275	0.152137	−	0.988359i	$-0.451384\pi$
$409$	6.15357	0.304275	0.152137	+	0.988359i	$0.451384\pi$
$410$	0	0
$411$	22.0786	1.08906
$412$	0	0
$413$	0.272803	0.0134237
$414$	0	0
$415$	17.6310	0.865471
$416$	0	0
$417$	1.95579	0.0957756
$418$	0	0
$419$	20.1239	0.983118	0.491559	−	0.870844i	$-0.336427\pi$
$419$	20.1239	0.983118	0.491559	+	0.870844i	$0.336427\pi$
$420$	0	0
$421$	18.2139	0.887691	0.443846	−	0.896103i	$-0.353614\pi$
$421$	18.2139	0.887691	0.443846	+	0.896103i	$0.353614\pi$
$422$	0	0
$423$	−5.31896	−0.258617
$424$	0	0
$425$	16.8123	0.815515
$426$	0	0
$427$	42.8125	2.07184
$428$	0	0
$429$	3.36832	0.162624
$430$	0	0
$431$	−20.4864	−0.986795	−0.493398	−	0.869804i	$-0.664245\pi$
$431$	−20.4864	−0.986795	−0.493398	+	0.869804i	$0.664245\pi$
$432$	0	0
$433$	20.7693	0.998111	0.499055	−	0.866570i	$-0.333680\pi$
$433$	20.7693	0.998111	0.499055	+	0.866570i	$0.333680\pi$
$434$	0	0
$435$	12.5562	0.602022
$436$	0	0
$437$	33.3612	1.59588
$438$	0	0
$439$	−12.5543	−0.599186	−0.299593	−	0.954067i	$-0.596851\pi$
$439$	−12.5543	−0.599186	−0.299593	+	0.954067i	$0.596851\pi$
$440$	0	0
$441$	10.0509	0.478617
$442$	0	0
$443$	−0.828246	−0.0393511	−0.0196756	−	0.999806i	$-0.506263\pi$
$443$	−0.828246	−0.0393511	−0.0196756	+	0.999806i	$0.506263\pi$
$444$	0	0
$445$	−10.9514	−0.519148
$446$	0	0
$447$	1.90601	0.0901510
$448$	0	0
$449$	18.6090	0.878213	0.439107	−	0.898435i	$-0.355295\pi$
$449$	18.6090	0.878213	0.439107	+	0.898435i	$0.355295\pi$
$450$	0	0
$451$	2.54514	0.119846
$452$	0	0
$453$	−6.47280	−0.304119
$454$	0	0
$455$	−9.46768	−0.443851
$456$	0	0
$457$	−25.2882	−1.18293	−0.591466	−	0.806330i	$-0.701451\pi$
$457$	−25.2882	−1.18293	−0.591466	+	0.806330i	$0.701451\pi$
$458$	0	0
$459$	7.89205	0.368369
$460$	0	0
$461$	31.4877	1.46653	0.733263	−	0.679945i	$-0.237996\pi$
$461$	31.4877	1.46653	0.733263	+	0.679945i	$0.237996\pi$
$462$	0	0
$463$	−17.3225	−0.805044	−0.402522	−	0.915410i	$-0.631866\pi$
$463$	−17.3225	−0.805044	−0.402522	+	0.915410i	$0.631866\pi$
$464$	0	0
$465$	−7.95452	−0.368882
$466$	0	0
$467$	25.6138	1.18527	0.592633	−	0.805473i	$-0.298089\pi$
$467$	25.6138	1.18527	0.592633	+	0.805473i	$0.298089\pi$
$468$	0	0
$469$	−25.2148	−1.16431
$470$	0	0
$471$	−0.438526	−0.0202062
$472$	0	0
$473$	−21.6007	−0.993202
$474$	0	0
$475$	10.8364	0.497210
$476$	0	0
$477$	12.1374	0.555734
$478$	0	0
$479$	−19.1003	−0.872717	−0.436359	−	0.899773i	$-0.643732\pi$
$479$	−19.1003	−0.872717	−0.436359	+	0.899773i	$0.643732\pi$
$480$	0	0
$481$	−0.0180997	−0.000825276 0
$482$	0	0
$483$	27.0810	1.23223
$484$	0	0
$485$	−31.8123	−1.44452
$486$	0	0
$487$	−15.8243	−0.717068	−0.358534	−	0.933517i	$-0.616723\pi$
$487$	−15.8243	−0.717068	−0.358534	+	0.933517i	$0.616723\pi$
$488$	0	0
$489$	−12.2469	−0.553825
$490$	0	0
$491$	14.8932	0.672121	0.336061	−	0.941840i	$-0.390905\pi$
$491$	14.8932	0.672121	0.336061	+	0.941840i	$0.390905\pi$
$492$	0	0
$493$	−58.4961	−2.63453
$494$	0	0
$495$	4.21584	0.189488
$496$	0	0
$497$	41.1241	1.84467
$498$	0	0
$499$	5.95799	0.266716	0.133358	−	0.991068i	$-0.457424\pi$
$499$	5.95799	0.266716	0.133358	+	0.991068i	$0.457424\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	31.4302	1.40140	0.700701	−	0.713455i	$-0.252871\pi$
$503$	31.4302	1.40140	0.700701	+	0.713455i	$0.252871\pi$
$504$	0	0
$505$	−23.8179	−1.05988
$506$	0	0
$507$	11.1681	0.495993
$508$	0	0
$509$	−4.98258	−0.220849	−0.110424	−	0.993885i	$-0.535221\pi$
$509$	−4.98258	−0.220849	−0.110424	+	0.993885i	$0.535221\pi$
$510$	0	0
$511$	−15.5332	−0.687150
$512$	0	0
$513$	5.08686	0.224591
$514$	0	0
$515$	−30.2852	−1.33453
$516$	0	0
$517$	13.2371	0.582165
$518$	0	0
$519$	−8.71361	−0.382485
$520$	0	0
$521$	33.7533	1.47876	0.739379	−	0.673289i	$-0.235119\pi$
$521$	33.7533	1.47876	0.739379	+	0.673289i	$0.235119\pi$
$522$	0	0
$523$	−43.7117	−1.91138	−0.955690	−	0.294376i	$-0.904888\pi$
$523$	−43.7117	−1.91138	−0.955690	+	0.294376i	$0.904888\pi$
$524$	0	0
$525$	8.79652	0.383912
$526$	0	0
$527$	37.0582	1.61428
$528$	0	0
$529$	20.0112	0.870053
$530$	0	0
$531$	0.0660655	0.00286700
$532$	0	0
$533$	−1.38419	−0.0599561
$534$	0	0
$535$	22.2415	0.961583
$536$	0	0
$537$	21.3884	0.922977
$538$	0	0
$539$	−25.0133	−1.07740
$540$	0	0
$541$	−24.6423	−1.05946	−0.529728	−	0.848168i	$-0.677706\pi$
$541$	−24.6423	−1.05946	−0.529728	+	0.848168i	$0.677706\pi$
$542$	0	0
$543$	1.85644	0.0796676
$544$	0	0
$545$	−14.7557	−0.632066
$546$	0	0
$547$	17.1292	0.732390	0.366195	−	0.930538i	$-0.380660\pi$
$547$	17.1292	0.732390	0.366195	+	0.930538i	$0.380660\pi$
$548$	0	0
$549$	10.3680	0.442497
$550$	0	0
$551$	−37.7040	−1.60624
$552$	0	0
$553$	−47.1733	−2.00601
$554$	0	0
$555$	−0.0226538	−0.000961602 0
$556$	0	0
$557$	4.33857	0.183831	0.0919155	−	0.995767i	$-0.470701\pi$
$557$	4.33857	0.183831	0.0919155	+	0.995767i	$0.470701\pi$
$558$	0	0
$559$	11.7477	0.496875
$560$	0	0
$561$	−19.6406	−0.829226
$562$	0	0
$563$	0.824504	0.0347487	0.0173744	−	0.999849i	$-0.494469\pi$
$563$	0.824504	0.0347487	0.0173744	+	0.999849i	$0.494469\pi$
$564$	0	0
$565$	5.50580	0.231631
$566$	0	0
$567$	4.12928	0.173413
$568$	0	0
$569$	9.12359	0.382481	0.191240	−	0.981543i	$-0.438749\pi$
$569$	9.12359	0.382481	0.191240	+	0.981543i	$0.438749\pi$
$570$	0	0
$571$	6.28710	0.263107	0.131553	−	0.991309i	$-0.458003\pi$
$571$	6.28710	0.263107	0.131553	+	0.991309i	$0.458003\pi$
$572$	0	0
$573$	11.4811	0.479632
$574$	0	0
$575$	13.9710	0.582631
$576$	0	0
$577$	16.0704	0.669020	0.334510	−	0.942392i	$-0.391429\pi$
$577$	16.0704	0.669020	0.334510	+	0.942392i	$0.391429\pi$
$578$	0	0
$579$	25.0096	1.03936
$580$	0	0
$581$	−42.9765	−1.78296
$582$	0	0
$583$	−30.2058	−1.25100
$584$	0	0
$585$	−2.29282	−0.0947962
$586$	0	0
$587$	8.45147	0.348829	0.174415	−	0.984672i	$-0.444197\pi$
$587$	8.45147	0.348829	0.174415	+	0.984672i	$0.444197\pi$
$588$	0	0
$589$	23.8861	0.984208
$590$	0	0
$591$	−25.2259	−1.03765
$592$	0	0
$593$	31.4745	1.29250	0.646250	−	0.763125i	$-0.276336\pi$
$593$	31.4745	1.29250	0.646250	+	0.763125i	$0.276336\pi$
$594$	0	0
$595$	55.2057	2.26321
$596$	0	0
$597$	−25.2035	−1.03151
$598$	0	0
$599$	4.26080	0.174091	0.0870457	−	0.996204i	$-0.472257\pi$
$599$	4.26080	0.174091	0.0870457	+	0.996204i	$0.472257\pi$
$600$	0	0
$601$	−33.6592	−1.37299	−0.686493	−	0.727136i	$-0.740851\pi$
$601$	−33.6592	−1.37299	−0.686493	+	0.727136i	$0.740851\pi$
$602$	0	0
$603$	−6.10634	−0.248669
$604$	0	0
$605$	8.14250	0.331040
$606$	0	0
$607$	42.0428	1.70647	0.853233	−	0.521530i	$-0.174639\pi$
$607$	42.0428	1.70647	0.853233	+	0.521530i	$0.174639\pi$
$608$	0	0
$609$	−30.6063	−1.24023
$610$	0	0
$611$	−7.19907	−0.291243
$612$	0	0
$613$	40.1736	1.62260	0.811298	−	0.584632i	$-0.198761\pi$
$613$	40.1736	1.62260	0.811298	+	0.584632i	$0.198761\pi$
$614$	0	0
$615$	−1.73248	−0.0698602
$616$	0	0
$617$	−2.71175	−0.109171	−0.0545855	−	0.998509i	$-0.517384\pi$
$617$	−2.71175	−0.109171	−0.0545855	+	0.998509i	$0.517384\pi$
$618$	0	0
$619$	19.3649	0.778340	0.389170	−	0.921166i	$-0.372762\pi$
$619$	19.3649	0.778340	0.389170	+	0.921166i	$0.372762\pi$
$620$	0	0
$621$	6.55829	0.263175
$622$	0	0
$623$	26.6948	1.06950
$624$	0	0
$625$	−9.81051	−0.392420
$626$	0	0
$627$	−12.6594	−0.505570
$628$	0	0
$629$	0.105539	0.00420811
$630$	0	0
$631$	−31.9758	−1.27294	−0.636469	−	0.771302i	$-0.719606\pi$
$631$	−31.9758	−1.27294	−0.636469	+	0.771302i	$0.719606\pi$
$632$	0	0
$633$	−5.16157	−0.205154
$634$	0	0
$635$	−28.2846	−1.12244
$636$	0	0
$637$	13.6037	0.538997
$638$	0	0
$639$	9.95916	0.393978
$640$	0	0
$641$	−33.4659	−1.32182	−0.660912	−	0.750464i	$-0.729830\pi$
$641$	−33.4659	−1.32182	−0.660912	+	0.750464i	$0.729830\pi$
$642$	0	0
$643$	17.3135	0.682777	0.341388	−	0.939922i	$-0.389103\pi$
$643$	17.3135	0.682777	0.341388	+	0.939922i	$0.389103\pi$
$644$	0	0
$645$	14.7036	0.578953
$646$	0	0
$647$	−25.6129	−1.00695	−0.503473	−	0.864011i	$-0.667945\pi$
$647$	−25.6129	−1.00695	−0.503473	+	0.864011i	$0.667945\pi$
$648$	0	0
$649$	−0.164414	−0.00645382
$650$	0	0
$651$	19.3896	0.759938
$652$	0	0
$653$	9.35684	0.366161	0.183081	−	0.983098i	$-0.441393\pi$
$653$	9.35684	0.366161	0.183081	+	0.983098i	$0.441393\pi$
$654$	0	0
$655$	−15.7962	−0.617208
$656$	0	0
$657$	−3.76173	−0.146759
$658$	0	0
$659$	−1.42266	−0.0554188	−0.0277094	−	0.999616i	$-0.508821\pi$
$659$	−1.42266	−0.0554188	−0.0277094	+	0.999616i	$0.508821\pi$
$660$	0	0
$661$	4.25492	0.165497	0.0827485	−	0.996570i	$-0.473630\pi$
$661$	4.25492	0.165497	0.0827485	+	0.996570i	$0.473630\pi$
$662$	0	0
$663$	10.6817	0.414842
$664$	0	0
$665$	35.5831	1.37985
$666$	0	0
$667$	−48.6103	−1.88220
$668$	0	0
$669$	2.36856	0.0915738
$670$	0	0
$671$	−25.8025	−0.996093
$672$	0	0
$673$	−38.0475	−1.46663	−0.733313	−	0.679892i	$-0.762027\pi$
$673$	−38.0475	−1.46663	−0.733313	+	0.679892i	$0.762027\pi$
$674$	0	0
$675$	2.13028	0.0819945
$676$	0	0
$677$	−38.7193	−1.48810	−0.744052	−	0.668122i	$-0.767098\pi$
$677$	−38.7193	−1.48810	−0.744052	+	0.668122i	$0.767098\pi$
$678$	0	0
$679$	77.5442	2.97587
$680$	0	0
$681$	−0.662698	−0.0253946
$682$	0	0
$683$	8.56755	0.327828	0.163914	−	0.986475i	$-0.447588\pi$
$683$	8.56755	0.327828	0.163914	+	0.986475i	$0.447588\pi$
$684$	0	0
$685$	37.4017	1.42904
$686$	0	0
$687$	−28.4020	−1.08360
$688$	0	0
$689$	16.4277	0.625844
$690$	0	0
$691$	−31.7047	−1.20610	−0.603051	−	0.797702i	$-0.706049\pi$
$691$	−31.7047	−1.20610	−0.603051	+	0.797702i	$0.706049\pi$
$692$	0	0
$693$	−10.2764	−0.390366
$694$	0	0
$695$	3.31316	0.125675
$696$	0	0
$697$	8.07118	0.305718
$698$	0	0
$699$	−1.11158	−0.0420437
$700$	0	0
$701$	−43.0946	−1.62766	−0.813830	−	0.581103i	$-0.802621\pi$
$701$	−43.0946	−1.62766	−0.813830	+	0.581103i	$0.802621\pi$
$702$	0	0
$703$	0.0680256	0.00256563
$704$	0	0
$705$	−9.01045	−0.339353
$706$	0	0
$707$	58.0574	2.18347
$708$	0	0
$709$	−23.8750	−0.896643	−0.448321	−	0.893872i	$-0.647978\pi$
$709$	−23.8750	−0.896643	−0.448321	+	0.893872i	$0.647978\pi$
$710$	0	0
$711$	−11.4241	−0.428437
$712$	0	0
$713$	30.7954	1.15330
$714$	0	0
$715$	5.70603	0.213393
$716$	0	0
$717$	−5.03519	−0.188043
$718$	0	0
$719$	10.9206	0.407269	0.203635	−	0.979047i	$-0.434725\pi$
$719$	10.9206	0.407269	0.203635	+	0.979047i	$0.434725\pi$
$720$	0	0
$721$	73.8219	2.74927
$722$	0	0
$723$	−15.9384	−0.592755
$724$	0	0
$725$	−15.7897	−0.586415
$726$	0	0
$727$	15.2704	0.566348	0.283174	−	0.959069i	$-0.408613\pi$
$727$	15.2704	0.566348	0.283174	+	0.959069i	$0.408613\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−68.5005	−2.53358
$732$	0	0
$733$	21.4949	0.793931	0.396966	−	0.917834i	$-0.370063\pi$
$733$	21.4949	0.793931	0.396966	+	0.917834i	$0.370063\pi$
$734$	0	0
$735$	17.0266	0.628034
$736$	0	0
$737$	15.1966	0.559773
$738$	0	0
$739$	−24.6120	−0.905368	−0.452684	−	0.891671i	$-0.649533\pi$
$739$	−24.6120	−0.905368	−0.452684	+	0.891671i	$0.649533\pi$
$740$	0	0
$741$	6.88493	0.252924
$742$	0	0
$743$	29.9574	1.09903	0.549514	−	0.835484i	$-0.314813\pi$
$743$	29.9574	1.09903	0.549514	+	0.835484i	$0.314813\pi$
$744$	0	0
$745$	3.22882	0.118295
$746$	0	0
$747$	−10.4077	−0.380799
$748$	0	0
$749$	−54.2148	−1.98097
$750$	0	0
$751$	−33.9459	−1.23870	−0.619351	−	0.785114i	$-0.712604\pi$
$751$	−33.9459	−1.23870	−0.619351	+	0.785114i	$0.712604\pi$
$752$	0	0
$753$	12.4875	0.455070
$754$	0	0
$755$	−10.9651	−0.399060
$756$	0	0
$757$	13.7652	0.500303	0.250152	−	0.968207i	$-0.419520\pi$
$757$	13.7652	0.500303	0.250152	+	0.968207i	$0.419520\pi$
$758$	0	0
$759$	−16.3213	−0.592427
$760$	0	0
$761$	9.19742	0.333406	0.166703	−	0.986007i	$-0.446688\pi$
$761$	9.19742	0.333406	0.166703	+	0.986007i	$0.446688\pi$
$762$	0	0
$763$	35.9679	1.30213
$764$	0	0
$765$	13.3693	0.483369
$766$	0	0
$767$	0.0894178	0.00322869
$768$	0	0
$769$	23.3873	0.843368	0.421684	−	0.906743i	$-0.361439\pi$
$769$	23.3873	0.843368	0.421684	+	0.906743i	$0.361439\pi$
$770$	0	0
$771$	−13.4024	−0.482674
$772$	0	0
$773$	2.59137	0.0932050	0.0466025	−	0.998914i	$-0.485161\pi$
$773$	2.59137	0.0932050	0.0466025	+	0.998914i	$0.485161\pi$
$774$	0	0
$775$	10.0030	0.359319
$776$	0	0
$777$	0.0552200	0.00198101
$778$	0	0
$779$	5.20233	0.186393
$780$	0	0
$781$	−24.7849	−0.886873
$782$	0	0
$783$	−7.41203	−0.264884
$784$	0	0
$785$	−0.742874	−0.0265143
$786$	0	0
$787$	−7.99368	−0.284944	−0.142472	−	0.989799i	$-0.545505\pi$
$787$	−7.99368	−0.284944	−0.142472	+	0.989799i	$0.545505\pi$
$788$	0	0
$789$	10.2710	0.365657
$790$	0	0
$791$	−13.4207	−0.477185
$792$	0	0
$793$	14.0329	0.498321
$794$	0	0
$795$	20.5611	0.729227
$796$	0	0
$797$	15.4523	0.547348	0.273674	−	0.961823i	$-0.411761\pi$
$797$	15.4523	0.547348	0.273674	+	0.961823i	$0.411761\pi$
$798$	0	0
$799$	41.9775	1.48506
$800$	0	0
$801$	6.46475	0.228421
$802$	0	0
$803$	9.36165	0.330365
$804$	0	0
$805$	45.8759	1.61691
$806$	0	0
$807$	−21.0795	−0.742032
$808$	0	0
$809$	−8.25639	−0.290279	−0.145140	−	0.989411i	$-0.546363\pi$
$809$	−8.25639	−0.290279	−0.145140	+	0.989411i	$0.546363\pi$
$810$	0	0
$811$	26.6623	0.936240	0.468120	−	0.883665i	$-0.344932\pi$
$811$	26.6623	0.936240	0.468120	+	0.883665i	$0.344932\pi$
$812$	0	0
$813$	−22.7180	−0.796756
$814$	0	0
$815$	−20.7466	−0.726721
$816$	0	0
$817$	−44.1523	−1.54470
$818$	0	0
$819$	5.58886	0.195291
$820$	0	0
$821$	−7.60659	−0.265472	−0.132736	−	0.991151i	$-0.542376\pi$
$821$	−7.60659	−0.265472	−0.132736	+	0.991151i	$0.542376\pi$
$822$	0	0
$823$	−33.8995	−1.18166	−0.590832	−	0.806795i	$-0.701200\pi$
$823$	−33.8995	−1.18166	−0.590832	+	0.806795i	$0.701200\pi$
$824$	0	0
$825$	−5.30153	−0.184576
$826$	0	0
$827$	−22.3492	−0.777157	−0.388578	−	0.921416i	$-0.627034\pi$
$827$	−22.3492	−0.777157	−0.388578	+	0.921416i	$0.627034\pi$
$828$	0	0
$829$	15.8694	0.551166	0.275583	−	0.961277i	$-0.411129\pi$
$829$	15.8694	0.551166	0.275583	+	0.961277i	$0.411129\pi$
$830$	0	0
$831$	25.7334	0.892681
$832$	0	0
$833$	−79.3226	−2.74836
$834$	0	0
$835$	1.69402	0.0586241
$836$	0	0
$837$	4.69564	0.162305
$838$	0	0
$839$	−7.54100	−0.260344	−0.130172	−	0.991491i	$-0.541553\pi$
$839$	−7.54100	−0.260344	−0.130172	+	0.991491i	$0.541553\pi$
$840$	0	0
$841$	25.9382	0.894419
$842$	0	0
$843$	20.0242	0.689669
$844$	0	0
$845$	18.9191	0.650835
$846$	0	0
$847$	−19.8478	−0.681978
$848$	0	0
$849$	−7.01267	−0.240674
$850$	0	0
$851$	0.0877027	0.00300641
$852$	0	0
$853$	39.5679	1.35478	0.677389	−	0.735625i	$-0.263111\pi$
$853$	39.5679	1.35478	0.677389	+	0.735625i	$0.263111\pi$
$854$	0	0
$855$	8.61727	0.294705
$856$	0	0
$857$	23.3571	0.797864	0.398932	−	0.916981i	$-0.369381\pi$
$857$	23.3571	0.797864	0.398932	+	0.916981i	$0.369381\pi$
$858$	0	0
$859$	43.6810	1.49038	0.745189	−	0.666854i	$-0.232359\pi$
$859$	43.6810	1.49038	0.745189	+	0.666854i	$0.232359\pi$
$860$	0	0
$861$	4.22301	0.143920
$862$	0	0
$863$	41.6590	1.41809	0.709044	−	0.705165i	$-0.249127\pi$
$863$	41.6590	1.41809	0.709044	+	0.705165i	$0.249127\pi$
$864$	0	0
$865$	−14.7611	−0.501891
$866$	0	0
$867$	−45.2844	−1.53794
$868$	0	0
$869$	28.4306	0.964443
$870$	0	0
$871$	−8.26476	−0.280041
$872$	0	0
$873$	18.7791	0.635576
$874$	0	0
$875$	49.8770	1.68615
$876$	0	0
$877$	−44.4805	−1.50200	−0.751000	−	0.660302i	$-0.770428\pi$
$877$	−44.4805	−1.50200	−0.751000	+	0.660302i	$0.770428\pi$
$878$	0	0
$879$	−2.63359	−0.0888287
$880$	0	0
$881$	−4.85366	−0.163524	−0.0817619	−	0.996652i	$-0.526055\pi$
$881$	−4.85366	−0.163524	−0.0817619	+	0.996652i	$0.526055\pi$
$882$	0	0
$883$	−31.3649	−1.05551	−0.527757	−	0.849395i	$-0.676967\pi$
$883$	−31.3649	−1.05551	−0.527757	+	0.849395i	$0.676967\pi$
$884$	0	0
$885$	0.111917	0.00376203
$886$	0	0
$887$	14.4960	0.486727	0.243364	−	0.969935i	$-0.421749\pi$
$887$	14.4960	0.486727	0.243364	+	0.969935i	$0.421749\pi$
$888$	0	0
$889$	68.9454	2.31235
$890$	0	0
$891$	−2.48865	−0.0833731
$892$	0	0
$893$	27.0568	0.905423
$894$	0	0
$895$	36.2325	1.21112
$896$	0	0
$897$	8.87647	0.296377
$898$	0	0
$899$	−34.8042	−1.16078
$900$	0	0
$901$	−95.7891	−3.19120
$902$	0	0
$903$	−35.8408	−1.19271
$904$	0	0
$905$	3.14486	0.104539
$906$	0	0
$907$	−3.11429	−0.103408	−0.0517042	−	0.998662i	$-0.516465\pi$
$907$	−3.11429	−0.103408	−0.0517042	+	0.998662i	$0.516465\pi$
$908$	0	0
$909$	14.0599	0.466338
$910$	0	0
$911$	56.8426	1.88328	0.941640	−	0.336620i	$-0.109284\pi$
$911$	56.8426	1.88328	0.941640	+	0.336620i	$0.109284\pi$
$912$	0	0
$913$	25.9013	0.857207
$914$	0	0
$915$	17.5637	0.580639
$916$	0	0
$917$	38.5041	1.27152
$918$	0	0
$919$	−35.0944	−1.15766	−0.578829	−	0.815449i	$-0.696490\pi$
$919$	−35.0944	−1.15766	−0.578829	+	0.815449i	$0.696490\pi$
$920$	0	0
$921$	12.3105	0.405645
$922$	0	0
$923$	13.4794	0.443681
$924$	0	0
$925$	0.0284878	0.000936673 0
$926$	0	0
$927$	17.8777	0.587180
$928$	0	0
$929$	−11.1328	−0.365255	−0.182627	−	0.983182i	$-0.558460\pi$
$929$	−11.1328	−0.365255	−0.182627	+	0.983182i	$0.558460\pi$
$930$	0	0
$931$	−51.1278	−1.67565
$932$	0	0
$933$	23.9999	0.785720
$934$	0	0
$935$	−33.2716	−1.08810
$936$	0	0
$937$	1.43915	0.0470149	0.0235075	−	0.999724i	$-0.492517\pi$
$937$	1.43915	0.0470149	0.0235075	+	0.999724i	$0.492517\pi$
$938$	0	0
$939$	−3.38113	−0.110339
$940$	0	0
$941$	−27.6688	−0.901976	−0.450988	−	0.892530i	$-0.648928\pi$
$941$	−27.6688	−0.901976	−0.450988	+	0.892530i	$0.648928\pi$
$942$	0	0
$943$	6.70715	0.218415
$944$	0	0
$945$	6.99510	0.227551
$946$	0	0
$947$	−22.1562	−0.719979	−0.359989	−	0.932956i	$-0.617220\pi$
$947$	−22.1562	−0.719979	−0.359989	+	0.932956i	$0.617220\pi$
$948$	0	0
$949$	−5.09140	−0.165274
$950$	0	0
$951$	7.85532	0.254726
$952$	0	0
$953$	18.1214	0.587010	0.293505	−	0.955957i	$-0.405178\pi$
$953$	18.1214	0.587010	0.293505	+	0.955957i	$0.405178\pi$
$954$	0	0
$955$	19.4493	0.629366
$956$	0	0
$957$	18.4460	0.596274
$958$	0	0
$959$	−91.1686	−2.94399
$960$	0	0
$961$	−8.95101	−0.288742
$962$	0	0
$963$	−13.1294	−0.423088
$964$	0	0
$965$	42.3669	1.36384
$966$	0	0
$967$	−40.2636	−1.29479	−0.647395	−	0.762155i	$-0.724142\pi$
$967$	−40.2636	−1.29479	−0.647395	+	0.762155i	$0.724142\pi$
$968$	0	0
$969$	−40.1458	−1.28967
$970$	0	0
$971$	−16.8192	−0.539752	−0.269876	−	0.962895i	$-0.586983\pi$
$971$	−16.8192	−0.539752	−0.269876	+	0.962895i	$0.586983\pi$
$972$	0	0
$973$	−8.07601	−0.258905
$974$	0	0
$975$	2.88327	0.0923387
$976$	0	0
$977$	18.1817	0.581685	0.290842	−	0.956771i	$-0.406064\pi$
$977$	18.1817	0.581685	0.290842	+	0.956771i	$0.406064\pi$
$978$	0	0
$979$	−16.0885	−0.514192
$980$	0	0
$981$	8.71046	0.278104
$982$	0	0
$983$	−38.6117	−1.23152	−0.615761	−	0.787933i	$-0.711151\pi$
$983$	−38.6117	−1.23152	−0.615761	+	0.787933i	$0.711151\pi$
$984$	0	0
$985$	−42.7332	−1.36159
$986$	0	0
$987$	21.9635	0.699105
$988$	0	0
$989$	−56.9239	−1.81007
$990$	0	0
$991$	−6.98885	−0.222008	−0.111004	−	0.993820i	$-0.535407\pi$
$991$	−6.98885	−0.222008	−0.111004	+	0.993820i	$0.535407\pi$
$992$	0	0
$993$	−24.1080	−0.765045
$994$	0	0
$995$	−42.6954	−1.35353
$996$	0	0
$997$	−5.03760	−0.159542	−0.0797712	−	0.996813i	$-0.525419\pi$
$997$	−5.03760	−0.159542	−0.0797712	+	0.996813i	$0.525419\pi$
$998$	0	0
$999$	0.0133728	0.000423097 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bb.1.2		9
4.3	odd	2		2004.2.a.d.1.2	✓	9
12.11	even	2		6012.2.a.h.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.2	✓	9	4.3	odd	2
6012.2.a.h.1.8		9	12.11	even	2
8016.2.a.bb.1.2		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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.69402 q^{5} +4.12928 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.69402 q^{5} +4.12928 q^{7} +1.00000 q^{9} -2.48865 q^{11} +1.35347 q^{13} +1.69402 q^{15} -7.89205 q^{17} -5.08686 q^{19} -4.12928 q^{21} -6.55829 q^{23} -2.13028 q^{25} -1.00000 q^{27} +7.41203 q^{29} -4.69564 q^{31} +2.48865 q^{33} -6.99510 q^{35} -0.0133728 q^{37} -1.35347 q^{39} -1.02270 q^{41} +8.67968 q^{43} -1.69402 q^{45} -5.31896 q^{47} +10.0509 q^{49} +7.89205 q^{51} +12.1374 q^{53} +4.21584 q^{55} +5.08686 q^{57} +0.0660655 q^{59} +10.3680 q^{61} +4.12928 q^{63} -2.29282 q^{65} -6.10634 q^{67} +6.55829 q^{69} +9.95916 q^{71} -3.76173 q^{73} +2.13028 q^{75} -10.2764 q^{77} -11.4241 q^{79} +1.00000 q^{81} -10.4077 q^{83} +13.3693 q^{85} -7.41203 q^{87} +6.46475 q^{89} +5.58886 q^{91} +4.69564 q^{93} +8.61727 q^{95} +18.7791 q^{97} -2.48865 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9} - 7 q^{11} + 6 q^{13} - 9 q^{15} + 7 q^{17} - 2 q^{19} + 2 q^{21} - 19 q^{23} + 22 q^{25} - 9 q^{27} + 13 q^{29} - 12 q^{31} + 7 q^{33} - 4 q^{35} + 15 q^{37} - 6 q^{39} + 18 q^{41} + 6 q^{43} + 9 q^{45} - 25 q^{47} + 19 q^{49} - 7 q^{51} + 17 q^{53} + 3 q^{55} + 2 q^{57} - 3 q^{59} + 14 q^{61} - 2 q^{63} + 14 q^{65} + 4 q^{67} + 19 q^{69} - 17 q^{71} - 20 q^{73} - 22 q^{75} + 14 q^{77} + 8 q^{79} + 9 q^{81} + q^{83} + 5 q^{85} - 13 q^{87} + 36 q^{89} + 41 q^{91} + 12 q^{93} - 5 q^{95} + 31 q^{97} - 7 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9 - 7 * q^11 + 6 * q^13 - 9 * q^15 + 7 * q^17 - 2 * q^19 + 2 * q^21 - 19 * q^23 + 22 * q^25 - 9 * q^27 + 13 * q^29 - 12 * q^31 + 7 * q^33 - 4 * q^35 + 15 * q^37 - 6 * q^39 + 18 * q^41 + 6 * q^43 + 9 * q^45 - 25 * q^47 + 19 * q^49 - 7 * q^51 + 17 * q^53 + 3 * q^55 + 2 * q^57 - 3 * q^59 + 14 * q^61 - 2 * q^63 + 14 * q^65 + 4 * q^67 + 19 * q^69 - 17 * q^71 - 20 * q^73 - 22 * q^75 + 14 * q^77 + 8 * q^79 + 9 * q^81 + q^83 + 5 * q^85 - 13 * q^87 + 36 * q^89 + 41 * q^91 + 12 * q^93 - 5 * q^95 + 31 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more