LMFDB - Embedded newform 8016.2.a.x.1.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 $ x^8 - 3x^7 - 8x^6 + 28x^5 + 9x^4 - 64x^3 + 17x^2 + 23*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.60389$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.122001 q^{5} -3.47013 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.122001 q^{5} -3.47013 q^{7} +1.00000 q^{9} -2.22605 q^{11} -1.33362 q^{13} +0.122001 q^{15} +5.50322 q^{17} +0.640187 q^{19} +3.47013 q^{21} +0.402422 q^{23} -4.98512 q^{25} -1.00000 q^{27} +0.479682 q^{29} +7.34794 q^{31} +2.22605 q^{33} +0.423358 q^{35} +4.15831 q^{37} +1.33362 q^{39} -5.86917 q^{41} +2.46909 q^{43} -0.122001 q^{45} +1.14491 q^{47} +5.04182 q^{49} -5.50322 q^{51} -1.96789 q^{53} +0.271580 q^{55} -0.640187 q^{57} +0.167733 q^{59} -8.97251 q^{61} -3.47013 q^{63} +0.162703 q^{65} +6.23491 q^{67} -0.402422 q^{69} -2.61717 q^{71} +4.65157 q^{73} +4.98512 q^{75} +7.72470 q^{77} +15.8128 q^{79} +1.00000 q^{81} +4.00231 q^{83} -0.671396 q^{85} -0.479682 q^{87} +9.95997 q^{89} +4.62784 q^{91} -7.34794 q^{93} -0.0781032 q^{95} -0.0188520 q^{97} -2.22605 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9	$ 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9 - 13 * q^11 - 7 * q^15 + 11 * q^17 - 12 * q^19 - 4 * q^21 - 7 * q^23 - 5 * q^25 - 8 * q^27 + q^29 + 2 * q^31 + 13 * q^33 + 4 * q^35 - 9 * q^37 + 4 * q^41 - 2 * q^43 + 7 * q^45 - 17 * q^47 - 2 * q^49 - 11 * q^51 + 9 * q^53 - 7 * q^55 + 12 * q^57 - 29 * q^59 - 12 * q^61 + 4 * q^63 + 8 * q^65 + 7 * q^69 - 13 * q^71 - 20 * q^73 + 5 * q^75 - 22 * q^77 - 8 * q^79 + 8 * q^81 - 33 * q^83 - 31 * q^85 - q^87 + 4 * q^89 - q^91 - 2 * q^93 - 3 * q^95 - 31 * q^97 - 13 * 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
$5$	−0.122001	−0.0545603	−0.0272802	−	0.999628i	$-0.508685\pi$
$5$	−0.122001	−0.0545603	−0.0272802	+	0.999628i	$0.508685\pi$
$6$	0	0
$7$	−3.47013	−1.31159	−0.655793	−	0.754940i	$-0.727666\pi$
$7$	−3.47013	−1.31159	−0.655793	+	0.754940i	$0.727666\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.22605	−0.671180	−0.335590	−	0.942008i	$-0.608936\pi$
$11$	−2.22605	−0.671180	−0.335590	+	0.942008i	$0.608936\pi$
$12$	0	0
$13$	−1.33362	−0.369880	−0.184940	−	0.982750i	$-0.559209\pi$
$13$	−1.33362	−0.369880	−0.184940	+	0.982750i	$0.559209\pi$
$14$	0	0
$15$	0.122001	0.0315004
$16$	0	0
$17$	5.50322	1.33473	0.667364	−	0.744732i	$-0.267423\pi$
$17$	5.50322	1.33473	0.667364	+	0.744732i	$0.267423\pi$
$18$	0	0
$19$	0.640187	0.146869	0.0734345	−	0.997300i	$-0.476604\pi$
$19$	0.640187	0.146869	0.0734345	+	0.997300i	$0.476604\pi$
$20$	0	0
$21$	3.47013	0.757245
$22$	0	0
$23$	0.402422	0.0839108	0.0419554	−	0.999119i	$-0.486641\pi$
$23$	0.402422	0.0839108	0.0419554	+	0.999119i	$0.486641\pi$
$24$	0	0
$25$	−4.98512	−0.997023
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.479682	0.0890747	0.0445374	−	0.999008i	$-0.485819\pi$
$29$	0.479682	0.0890747	0.0445374	+	0.999008i	$0.485819\pi$
$30$	0	0
$31$	7.34794	1.31973	0.659865	−	0.751384i	$-0.270614\pi$
$31$	7.34794	1.31973	0.659865	+	0.751384i	$0.270614\pi$
$32$	0	0
$33$	2.22605	0.387506
$34$	0	0
$35$	0.423358	0.0715606
$36$	0	0
$37$	4.15831	0.683622	0.341811	−	0.939769i	$-0.388960\pi$
$37$	4.15831	0.683622	0.341811	+	0.939769i	$0.388960\pi$
$38$	0	0
$39$	1.33362	0.213550
$40$	0	0
$41$	−5.86917	−0.916610	−0.458305	−	0.888795i	$-0.651543\pi$
$41$	−5.86917	−0.916610	−0.458305	+	0.888795i	$0.651543\pi$
$42$	0	0
$43$	2.46909	0.376533	0.188266	−	0.982118i	$-0.439713\pi$
$43$	2.46909	0.376533	0.188266	+	0.982118i	$0.439713\pi$
$44$	0	0
$45$	−0.122001	−0.0181868
$46$	0	0
$47$	1.14491	0.167002	0.0835011	−	0.996508i	$-0.473390\pi$
$47$	1.14491	0.167002	0.0835011	+	0.996508i	$0.473390\pi$
$48$	0	0
$49$	5.04182	0.720260
$50$	0	0
$51$	−5.50322	−0.770605
$52$	0	0
$53$	−1.96789	−0.270311	−0.135155	−	0.990824i	$-0.543153\pi$
$53$	−1.96789	−0.270311	−0.135155	+	0.990824i	$0.543153\pi$
$54$	0	0
$55$	0.271580	0.0366198
$56$	0	0
$57$	−0.640187	−0.0847949
$58$	0	0
$59$	0.167733	0.0218370	0.0109185	−	0.999940i	$-0.496524\pi$
$59$	0.167733	0.0218370	0.0109185	+	0.999940i	$0.496524\pi$
$60$	0	0
$61$	−8.97251	−1.14881	−0.574406	−	0.818571i	$-0.694767\pi$
$61$	−8.97251	−1.14881	−0.574406	+	0.818571i	$0.694767\pi$
$62$	0	0
$63$	−3.47013	−0.437196
$64$	0	0
$65$	0.162703	0.0201808
$66$	0	0
$67$	6.23491	0.761716	0.380858	−	0.924634i	$-0.375629\pi$
$67$	6.23491	0.761716	0.380858	+	0.924634i	$0.375629\pi$
$68$	0	0
$69$	−0.402422	−0.0484459
$70$	0	0
$71$	−2.61717	−0.310601	−0.155300	−	0.987867i	$-0.549635\pi$
$71$	−2.61717	−0.310601	−0.155300	+	0.987867i	$0.549635\pi$
$72$	0	0
$73$	4.65157	0.544425	0.272213	−	0.962237i	$-0.412245\pi$
$73$	4.65157	0.544425	0.272213	+	0.962237i	$0.412245\pi$
$74$	0	0
$75$	4.98512	0.575632
$76$	0	0
$77$	7.72470	0.880311
$78$	0	0
$79$	15.8128	1.77908	0.889540	−	0.456857i	$-0.151025\pi$
$79$	15.8128	1.77908	0.889540	+	0.456857i	$0.151025\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00231	0.439310	0.219655	−	0.975578i	$-0.429507\pi$
$83$	4.00231	0.439310	0.219655	+	0.975578i	$0.429507\pi$
$84$	0	0
$85$	−0.671396	−0.0728231
$86$	0	0
$87$	−0.479682	−0.0514273
$88$	0	0
$89$	9.95997	1.05575	0.527877	−	0.849321i	$-0.322988\pi$
$89$	9.95997	1.05575	0.527877	+	0.849321i	$0.322988\pi$
$90$	0	0
$91$	4.62784	0.485130
$92$	0	0
$93$	−7.34794	−0.761946
$94$	0	0
$95$	−0.0781032	−0.00801322
$96$	0	0
$97$	−0.0188520	−0.00191413	−0.000957066	−	1.00000i	$-0.500305\pi$
$97$	−0.0188520	−0.00191413	−0.000957066		1.00000i	$0.500305\pi$
$98$	0	0
$99$	−2.22605	−0.223727
$100$	0	0
$101$	1.69668	0.168826	0.0844128	−	0.996431i	$-0.473099\pi$
$101$	1.69668	0.168826	0.0844128	+	0.996431i	$0.473099\pi$
$102$	0	0
$103$	4.92211	0.484990	0.242495	−	0.970153i	$-0.422034\pi$
$103$	4.92211	0.484990	0.242495	+	0.970153i	$0.422034\pi$
$104$	0	0
$105$	−0.423358	−0.0413155
$106$	0	0
$107$	−7.18014	−0.694131	−0.347065	−	0.937841i	$-0.612822\pi$
$107$	−7.18014	−0.694131	−0.347065	+	0.937841i	$0.612822\pi$
$108$	0	0
$109$	7.07053	0.677234	0.338617	−	0.940924i	$-0.390041\pi$
$109$	7.07053	0.677234	0.338617	+	0.940924i	$0.390041\pi$
$110$	0	0
$111$	−4.15831	−0.394690
$112$	0	0
$113$	10.7537	1.01163	0.505814	−	0.862643i	$-0.331192\pi$
$113$	10.7537	1.01163	0.505814	+	0.862643i	$0.331192\pi$
$114$	0	0
$115$	−0.0490957	−0.00457820
$116$	0	0
$117$	−1.33362	−0.123293
$118$	0	0
$119$	−19.0969	−1.75061
$120$	0	0
$121$	−6.04469	−0.549518
$122$	0	0
$123$	5.86917	0.529205
$124$	0	0
$125$	1.21819	0.108958
$126$	0	0
$127$	−17.5746	−1.55949	−0.779745	−	0.626098i	$-0.784651\pi$
$127$	−17.5746	−1.55949	−0.779745	+	0.626098i	$0.784651\pi$
$128$	0	0
$129$	−2.46909	−0.217391
$130$	0	0
$131$	−18.6310	−1.62780	−0.813900	−	0.581005i	$-0.802660\pi$
$131$	−18.6310	−1.62780	−0.813900	+	0.581005i	$0.802660\pi$
$132$	0	0
$133$	−2.22153	−0.192631
$134$	0	0
$135$	0.122001	0.0105001
$136$	0	0
$137$	−20.8065	−1.77762	−0.888811	−	0.458273i	$-0.848468\pi$
$137$	−20.8065	−1.77762	−0.888811	+	0.458273i	$0.848468\pi$
$138$	0	0
$139$	−5.60933	−0.475778	−0.237889	−	0.971292i	$-0.576455\pi$
$139$	−5.60933	−0.475778	−0.237889	+	0.971292i	$0.576455\pi$
$140$	0	0
$141$	−1.14491	−0.0964187
$142$	0	0
$143$	2.96871	0.248256
$144$	0	0
$145$	−0.0585215	−0.00485994
$146$	0	0
$147$	−5.04182	−0.415843
$148$	0	0
$149$	−5.30839	−0.434881	−0.217440	−	0.976074i	$-0.569771\pi$
$149$	−5.30839	−0.434881	−0.217440	+	0.976074i	$0.569771\pi$
$150$	0	0
$151$	8.97040	0.730001	0.365001	−	0.931007i	$-0.381069\pi$
$151$	8.97040	0.730001	0.365001	+	0.931007i	$0.381069\pi$
$152$	0	0
$153$	5.50322	0.444909
$154$	0	0
$155$	−0.896453	−0.0720049
$156$	0	0
$157$	−6.11217	−0.487804	−0.243902	−	0.969800i	$-0.578428\pi$
$157$	−6.11217	−0.487804	−0.243902	+	0.969800i	$0.578428\pi$
$158$	0	0
$159$	1.96789	0.156064
$160$	0	0
$161$	−1.39646	−0.110056
$162$	0	0
$163$	−18.3263	−1.43542	−0.717712	−	0.696340i	$-0.754810\pi$
$163$	−18.3263	−1.43542	−0.717712	+	0.696340i	$0.754810\pi$
$164$	0	0
$165$	−0.271580	−0.0211424
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.2215	−0.863189
$170$	0	0
$171$	0.640187	0.0489563
$172$	0	0
$173$	8.47265	0.644163	0.322082	−	0.946712i	$-0.395617\pi$
$173$	8.47265	0.644163	0.322082	+	0.946712i	$0.395617\pi$
$174$	0	0
$175$	17.2990	1.30768
$176$	0	0
$177$	−0.167733	−0.0126076
$178$	0	0
$179$	−18.9516	−1.41651	−0.708253	−	0.705959i	$-0.750516\pi$
$179$	−18.9516	−1.41651	−0.708253	+	0.705959i	$0.750516\pi$
$180$	0	0
$181$	15.2822	1.13592	0.567958	−	0.823058i	$-0.307734\pi$
$181$	15.2822	1.13592	0.567958	+	0.823058i	$0.307734\pi$
$182$	0	0
$183$	8.97251	0.663267
$184$	0	0
$185$	−0.507316	−0.0372986
$186$	0	0
$187$	−12.2505	−0.895842
$188$	0	0
$189$	3.47013	0.252415
$190$	0	0
$191$	4.11452	0.297716	0.148858	−	0.988859i	$-0.452440\pi$
$191$	4.11452	0.297716	0.148858	+	0.988859i	$0.452440\pi$
$192$	0	0
$193$	−12.7711	−0.919286	−0.459643	−	0.888104i	$-0.652023\pi$
$193$	−12.7711	−0.919286	−0.459643	+	0.888104i	$0.652023\pi$
$194$	0	0
$195$	−0.162703	−0.0116514
$196$	0	0
$197$	22.5859	1.60918	0.804588	−	0.593833i	$-0.202386\pi$
$197$	22.5859	1.60918	0.804588	+	0.593833i	$0.202386\pi$
$198$	0	0
$199$	7.40505	0.524930	0.262465	−	0.964941i	$-0.415465\pi$
$199$	7.40505	0.524930	0.262465	+	0.964941i	$0.415465\pi$
$200$	0	0
$201$	−6.23491	−0.439777
$202$	0	0
$203$	−1.66456	−0.116829
$204$	0	0
$205$	0.716042	0.0500105
$206$	0	0
$207$	0.402422	0.0279703
$208$	0	0
$209$	−1.42509	−0.0985755
$210$	0	0
$211$	6.39991	0.440588	0.220294	−	0.975434i	$-0.429298\pi$
$211$	6.39991	0.440588	0.220294	+	0.975434i	$0.429298\pi$
$212$	0	0
$213$	2.61717	0.179325
$214$	0	0
$215$	−0.301230	−0.0205437
$216$	0	0
$217$	−25.4983	−1.73094
$218$	0	0
$219$	−4.65157	−0.314324
$220$	0	0
$221$	−7.33922	−0.493689
$222$	0	0
$223$	−2.33822	−0.156579	−0.0782893	−	0.996931i	$-0.524946\pi$
$223$	−2.33822	−0.156579	−0.0782893	+	0.996931i	$0.524946\pi$
$224$	0	0
$225$	−4.98512	−0.332341
$226$	0	0
$227$	−18.9294	−1.25639	−0.628195	−	0.778056i	$-0.716206\pi$
$227$	−18.9294	−1.25639	−0.628195	+	0.778056i	$0.716206\pi$
$228$	0	0
$229$	−18.8437	−1.24523	−0.622615	−	0.782529i	$-0.713930\pi$
$229$	−18.8437	−1.24523	−0.622615	+	0.782529i	$0.713930\pi$
$230$	0	0
$231$	−7.72470	−0.508248
$232$	0	0
$233$	7.00542	0.458940	0.229470	−	0.973316i	$-0.426301\pi$
$233$	7.00542	0.458940	0.229470	+	0.973316i	$0.426301\pi$
$234$	0	0
$235$	−0.139680	−0.00911169
$236$	0	0
$237$	−15.8128	−1.02715
$238$	0	0
$239$	−2.78325	−0.180033	−0.0900166	−	0.995940i	$-0.528692\pi$
$239$	−2.78325	−0.180033	−0.0900166	+	0.995940i	$0.528692\pi$
$240$	0	0
$241$	28.9139	1.86251	0.931254	−	0.364370i	$-0.118716\pi$
$241$	28.9139	1.86251	0.931254	+	0.364370i	$0.118716\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.615105	−0.0392976
$246$	0	0
$247$	−0.853767	−0.0543239
$248$	0	0
$249$	−4.00231	−0.253636
$250$	0	0
$251$	27.6992	1.74836	0.874179	−	0.485604i	$-0.161400\pi$
$251$	27.6992	1.74836	0.874179	+	0.485604i	$0.161400\pi$
$252$	0	0
$253$	−0.895812	−0.0563192
$254$	0	0
$255$	0.671396	0.0420445
$256$	0	0
$257$	7.07503	0.441328	0.220664	−	0.975350i	$-0.429178\pi$
$257$	7.07503	0.441328	0.220664	+	0.975350i	$0.429178\pi$
$258$	0	0
$259$	−14.4299	−0.896630
$260$	0	0
$261$	0.479682	0.0296916
$262$	0	0
$263$	−14.5697	−0.898405	−0.449203	−	0.893430i	$-0.648292\pi$
$263$	−14.5697	−0.898405	−0.449203	+	0.893430i	$0.648292\pi$
$264$	0	0
$265$	0.240084	0.0147482
$266$	0	0
$267$	−9.95997	−0.609540
$268$	0	0
$269$	−13.6707	−0.833514	−0.416757	−	0.909018i	$-0.636833\pi$
$269$	−13.6707	−0.833514	−0.416757	+	0.909018i	$0.636833\pi$
$270$	0	0
$271$	−19.4027	−1.17863	−0.589316	−	0.807903i	$-0.700603\pi$
$271$	−19.4027	−1.17863	−0.589316	+	0.807903i	$0.700603\pi$
$272$	0	0
$273$	−4.62784	−0.280090
$274$	0	0
$275$	11.0971	0.669182
$276$	0	0
$277$	−32.8515	−1.97386	−0.986929	−	0.161158i	$-0.948477\pi$
$277$	−32.8515	−1.97386	−0.986929	+	0.161158i	$0.948477\pi$
$278$	0	0
$279$	7.34794	0.439910
$280$	0	0
$281$	−24.2094	−1.44421	−0.722107	−	0.691782i	$-0.756826\pi$
$281$	−24.2094	−1.44421	−0.722107	+	0.691782i	$0.756826\pi$
$282$	0	0
$283$	−20.9337	−1.24438	−0.622189	−	0.782867i	$-0.713757\pi$
$283$	−20.9337	−1.24438	−0.622189	+	0.782867i	$0.713757\pi$
$284$	0	0
$285$	0.0781032	0.00462643
$286$	0	0
$287$	20.3668	1.20221
$288$	0	0
$289$	13.2855	0.781497
$290$	0	0
$291$	0.0188520	0.00110513
$292$	0	0
$293$	28.6411	1.67323	0.836615	−	0.547791i	$-0.184531\pi$
$293$	28.6411	1.67323	0.836615	+	0.547791i	$0.184531\pi$
$294$	0	0
$295$	−0.0204636	−0.00119143
$296$	0	0
$297$	2.22605	0.129169
$298$	0	0
$299$	−0.536679	−0.0310369
$300$	0	0
$301$	−8.56807	−0.493856
$302$	0	0
$303$	−1.69668	−0.0974715
$304$	0	0
$305$	1.09465	0.0626795
$306$	0	0
$307$	−21.0765	−1.20290	−0.601450	−	0.798911i	$-0.705410\pi$
$307$	−21.0765	−1.20290	−0.601450	+	0.798911i	$0.705410\pi$
$308$	0	0
$309$	−4.92211	−0.280009
$310$	0	0
$311$	−29.8062	−1.69015	−0.845077	−	0.534645i	$-0.820445\pi$
$311$	−29.8062	−1.69015	−0.845077	+	0.534645i	$0.820445\pi$
$312$	0	0
$313$	−16.9090	−0.955750	−0.477875	−	0.878428i	$-0.658593\pi$
$313$	−16.9090	−0.955750	−0.477875	+	0.878428i	$0.658593\pi$
$314$	0	0
$315$	0.423358	0.0238535
$316$	0	0
$317$	−4.23258	−0.237725	−0.118863	−	0.992911i	$-0.537925\pi$
$317$	−4.23258	−0.237725	−0.118863	+	0.992911i	$0.537925\pi$
$318$	0	0
$319$	−1.06780	−0.0597852
$320$	0	0
$321$	7.18014	0.400757
$322$	0	0
$323$	3.52309	0.196030
$324$	0	0
$325$	6.64826	0.368779
$326$	0	0
$327$	−7.07053	−0.391001
$328$	0	0
$329$	−3.97299	−0.219038
$330$	0	0
$331$	−14.9764	−0.823180	−0.411590	−	0.911369i	$-0.635026\pi$
$331$	−14.9764	−0.823180	−0.411590	+	0.911369i	$0.635026\pi$
$332$	0	0
$333$	4.15831	0.227874
$334$	0	0
$335$	−0.760663	−0.0415595
$336$	0	0
$337$	8.18551	0.445893	0.222947	−	0.974831i	$-0.428432\pi$
$337$	8.18551	0.445893	0.222947	+	0.974831i	$0.428432\pi$
$338$	0	0
$339$	−10.7537	−0.584063
$340$	0	0
$341$	−16.3569	−0.885776
$342$	0	0
$343$	6.79513	0.366903
$344$	0	0
$345$	0.0490957	0.00264322
$346$	0	0
$347$	−10.0987	−0.542129	−0.271065	−	0.962561i	$-0.587376\pi$
$347$	−10.0987	−0.542129	−0.271065	+	0.962561i	$0.587376\pi$
$348$	0	0
$349$	1.67532	0.0896779	0.0448390	−	0.998994i	$-0.485723\pi$
$349$	1.67532	0.0896779	0.0448390	+	0.998994i	$0.485723\pi$
$350$	0	0
$351$	1.33362	0.0711835
$352$	0	0
$353$	−7.49257	−0.398789	−0.199395	−	0.979919i	$-0.563898\pi$
$353$	−7.49257	−0.398789	−0.199395	+	0.979919i	$0.563898\pi$
$354$	0	0
$355$	0.319296	0.0169465
$356$	0	0
$357$	19.0969	1.01072
$358$	0	0
$359$	27.5078	1.45181	0.725903	−	0.687797i	$-0.241422\pi$
$359$	27.5078	1.45181	0.725903	+	0.687797i	$0.241422\pi$
$360$	0	0
$361$	−18.5902	−0.978429
$362$	0	0
$363$	6.04469	0.317264
$364$	0	0
$365$	−0.567494	−0.0297040
$366$	0	0
$367$	−13.7190	−0.716125	−0.358063	−	0.933698i	$-0.616563\pi$
$367$	−13.7190	−0.716125	−0.358063	+	0.933698i	$0.616563\pi$
$368$	0	0
$369$	−5.86917	−0.305537
$370$	0	0
$371$	6.82885	0.354536
$372$	0	0
$373$	−5.17722	−0.268066	−0.134033	−	0.990977i	$-0.542793\pi$
$373$	−5.17722	−0.268066	−0.134033	+	0.990977i	$0.542793\pi$
$374$	0	0
$375$	−1.21819	−0.0629070
$376$	0	0
$377$	−0.639714	−0.0329470
$378$	0	0
$379$	−31.4919	−1.61763	−0.808816	−	0.588062i	$-0.799891\pi$
$379$	−31.4919	−1.61763	−0.808816	+	0.588062i	$0.799891\pi$
$380$	0	0
$381$	17.5746	0.900372
$382$	0	0
$383$	5.04367	0.257720	0.128860	−	0.991663i	$-0.458868\pi$
$383$	5.04367	0.257720	0.128860	+	0.991663i	$0.458868\pi$
$384$	0	0
$385$	−0.942417	−0.0480300
$386$	0	0
$387$	2.46909	0.125511
$388$	0	0
$389$	15.2976	0.775619	0.387810	−	0.921739i	$-0.373232\pi$
$389$	15.2976	0.775619	0.387810	+	0.921739i	$0.373232\pi$
$390$	0	0
$391$	2.21462	0.111998
$392$	0	0
$393$	18.6310	0.939811
$394$	0	0
$395$	−1.92917	−0.0970672
$396$	0	0
$397$	13.0851	0.656722	0.328361	−	0.944552i	$-0.393504\pi$
$397$	13.0851	0.656722	0.328361	+	0.944552i	$0.393504\pi$
$398$	0	0
$399$	2.22153	0.111216
$400$	0	0
$401$	11.5303	0.575797	0.287898	−	0.957661i	$-0.407043\pi$
$401$	11.5303	0.575797	0.287898	+	0.957661i	$0.407043\pi$
$402$	0	0
$403$	−9.79938	−0.488142
$404$	0	0
$405$	−0.122001	−0.00606226
$406$	0	0
$407$	−9.25662	−0.458834
$408$	0	0
$409$	4.50224	0.222621	0.111311	−	0.993786i	$-0.464495\pi$
$409$	4.50224	0.222621	0.111311	+	0.993786i	$0.464495\pi$
$410$	0	0
$411$	20.8065	1.02631
$412$	0	0
$413$	−0.582057	−0.0286411
$414$	0	0
$415$	−0.488284	−0.0239689
$416$	0	0
$417$	5.60933	0.274690
$418$	0	0
$419$	6.74123	0.329331	0.164665	−	0.986349i	$-0.447346\pi$
$419$	6.74123	0.329331	0.164665	+	0.986349i	$0.447346\pi$
$420$	0	0
$421$	−22.1305	−1.07857	−0.539286	−	0.842122i	$-0.681306\pi$
$421$	−22.1305	−1.07857	−0.539286	+	0.842122i	$0.681306\pi$
$422$	0	0
$423$	1.14491	0.0556674
$424$	0	0
$425$	−27.4342	−1.33075
$426$	0	0
$427$	31.1358	1.50677
$428$	0	0
$429$	−2.96871	−0.143331
$430$	0	0
$431$	16.7776	0.808150	0.404075	−	0.914726i	$-0.367593\pi$
$431$	16.7776	0.808150	0.404075	+	0.914726i	$0.367593\pi$
$432$	0	0
$433$	−21.3244	−1.02479	−0.512393	−	0.858751i	$-0.671241\pi$
$433$	−21.3244	−1.02479	−0.512393	+	0.858751i	$0.671241\pi$
$434$	0	0
$435$	0.0585215	0.00280589
$436$	0	0
$437$	0.257625	0.0123239
$438$	0	0
$439$	−7.37406	−0.351944	−0.175972	−	0.984395i	$-0.556307\pi$
$439$	−7.37406	−0.351944	−0.175972	+	0.984395i	$0.556307\pi$
$440$	0	0
$441$	5.04182	0.240087
$442$	0	0
$443$	16.5305	0.785386	0.392693	−	0.919670i	$-0.371544\pi$
$443$	16.5305	0.785386	0.392693	+	0.919670i	$0.371544\pi$
$444$	0	0
$445$	−1.21512	−0.0576023
$446$	0	0
$447$	5.30839	0.251078
$448$	0	0
$449$	−26.1807	−1.23554	−0.617772	−	0.786358i	$-0.711964\pi$
$449$	−26.1807	−1.23554	−0.617772	+	0.786358i	$0.711964\pi$
$450$	0	0
$451$	13.0651	0.615210
$452$	0	0
$453$	−8.97040	−0.421466
$454$	0	0
$455$	−0.564600	−0.0264688
$456$	0	0
$457$	14.4704	0.676896	0.338448	−	0.940985i	$-0.390098\pi$
$457$	14.4704	0.676896	0.338448	+	0.940985i	$0.390098\pi$
$458$	0	0
$459$	−5.50322	−0.256868
$460$	0	0
$461$	−38.2655	−1.78220	−0.891101	−	0.453805i	$-0.850066\pi$
$461$	−38.2655	−1.78220	−0.891101	+	0.453805i	$0.850066\pi$
$462$	0	0
$463$	38.1563	1.77327	0.886637	−	0.462466i	$-0.153035\pi$
$463$	38.1563	1.77327	0.886637	+	0.462466i	$0.153035\pi$
$464$	0	0
$465$	0.896453	0.0415720
$466$	0	0
$467$	31.3271	1.44964	0.724822	−	0.688936i	$-0.241922\pi$
$467$	31.3271	1.44964	0.724822	+	0.688936i	$0.241922\pi$
$468$	0	0
$469$	−21.6360	−0.999057
$470$	0	0
$471$	6.11217	0.281634
$472$	0	0
$473$	−5.49632	−0.252721
$474$	0	0
$475$	−3.19141	−0.146432
$476$	0	0
$477$	−1.96789	−0.0901036
$478$	0	0
$479$	−30.9290	−1.41318	−0.706591	−	0.707622i	$-0.749768\pi$
$479$	−30.9290	−1.41318	−0.706591	+	0.707622i	$0.749768\pi$
$480$	0	0
$481$	−5.54562	−0.252858
$482$	0	0
$483$	1.39646	0.0635410
$484$	0	0
$485$	0.00229996	0.000104436 0
$486$	0	0
$487$	2.26382	0.102583	0.0512917	−	0.998684i	$-0.483666\pi$
$487$	2.26382	0.102583	0.0512917	+	0.998684i	$0.483666\pi$
$488$	0	0
$489$	18.3263	0.828742
$490$	0	0
$491$	38.0911	1.71903	0.859513	−	0.511114i	$-0.170767\pi$
$491$	38.0911	1.71903	0.859513	+	0.511114i	$0.170767\pi$
$492$	0	0
$493$	2.63980	0.118890
$494$	0	0
$495$	0.271580	0.0122066
$496$	0	0
$497$	9.08192	0.407380
$498$	0	0
$499$	−21.3526	−0.955874	−0.477937	−	0.878394i	$-0.658615\pi$
$499$	−21.3526	−0.955874	−0.477937	+	0.878394i	$0.658615\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−6.58280	−0.293513	−0.146756	−	0.989173i	$-0.546883\pi$
$503$	−6.58280	−0.293513	−0.146756	+	0.989173i	$0.546883\pi$
$504$	0	0
$505$	−0.206996	−0.00921118
$506$	0	0
$507$	11.2215	0.498362
$508$	0	0
$509$	−24.8605	−1.10192	−0.550962	−	0.834531i	$-0.685739\pi$
$509$	−24.8605	−1.10192	−0.550962	+	0.834531i	$0.685739\pi$
$510$	0	0
$511$	−16.1416	−0.714061
$512$	0	0
$513$	−0.640187	−0.0282650
$514$	0	0
$515$	−0.600500	−0.0264612
$516$	0	0
$517$	−2.54863	−0.112088
$518$	0	0
$519$	−8.47265	−0.371908
$520$	0	0
$521$	6.71262	0.294085	0.147043	−	0.989130i	$-0.453025\pi$
$521$	6.71262	0.294085	0.147043	+	0.989130i	$0.453025\pi$
$522$	0	0
$523$	9.58608	0.419170	0.209585	−	0.977790i	$-0.432789\pi$
$523$	9.58608	0.419170	0.209585	+	0.977790i	$0.432789\pi$
$524$	0	0
$525$	−17.2990	−0.754991
$526$	0	0
$527$	40.4374	1.76148
$528$	0	0
$529$	−22.8381	−0.992959
$530$	0	0
$531$	0.167733	0.00727900
$532$	0	0
$533$	7.82725	0.339036
$534$	0	0
$535$	0.875982	0.0378720
$536$	0	0
$537$	18.9516	0.817820
$538$	0	0
$539$	−11.2234	−0.483424
$540$	0	0
$541$	−28.6874	−1.23337	−0.616683	−	0.787212i	$-0.711524\pi$
$541$	−28.6874	−1.23337	−0.616683	+	0.787212i	$0.711524\pi$
$542$	0	0
$543$	−15.2822	−0.655821
$544$	0	0
$545$	−0.862609	−0.0369501
$546$	0	0
$547$	21.1957	0.906263	0.453132	−	0.891444i	$-0.350307\pi$
$547$	21.1957	0.906263	0.453132	+	0.891444i	$0.350307\pi$
$548$	0	0
$549$	−8.97251	−0.382937
$550$	0	0
$551$	0.307086	0.0130823
$552$	0	0
$553$	−54.8726	−2.33342
$554$	0	0
$555$	0.507316	0.0215344
$556$	0	0
$557$	−14.1038	−0.597596	−0.298798	−	0.954316i	$-0.596586\pi$
$557$	−14.1038	−0.597596	−0.298798	+	0.954316i	$0.596586\pi$
$558$	0	0
$559$	−3.29283	−0.139272
$560$	0	0
$561$	12.2505	0.517215
$562$	0	0
$563$	−18.3618	−0.773857	−0.386928	−	0.922110i	$-0.626464\pi$
$563$	−18.3618	−0.773857	−0.386928	+	0.922110i	$0.626464\pi$
$564$	0	0
$565$	−1.31196	−0.0551947
$566$	0	0
$567$	−3.47013	−0.145732
$568$	0	0
$569$	34.0856	1.42894	0.714472	−	0.699664i	$-0.246667\pi$
$569$	34.0856	1.42894	0.714472	+	0.699664i	$0.246667\pi$
$570$	0	0
$571$	−23.1155	−0.967355	−0.483678	−	0.875246i	$-0.660699\pi$
$571$	−23.1155	−0.967355	−0.483678	+	0.875246i	$0.660699\pi$
$572$	0	0
$573$	−4.11452	−0.171887
$574$	0	0
$575$	−2.00612	−0.0836610
$576$	0	0
$577$	10.5995	0.441261	0.220631	−	0.975357i	$-0.429188\pi$
$577$	10.5995	0.441261	0.220631	+	0.975357i	$0.429188\pi$
$578$	0	0
$579$	12.7711	0.530750
$580$	0	0
$581$	−13.8885	−0.576194
$582$	0	0
$583$	4.38063	0.181427
$584$	0	0
$585$	0.162703	0.00672692
$586$	0	0
$587$	−28.8512	−1.19082	−0.595408	−	0.803423i	$-0.703010\pi$
$587$	−28.8512	−1.19082	−0.595408	+	0.803423i	$0.703010\pi$
$588$	0	0
$589$	4.70406	0.193827
$590$	0	0
$591$	−22.5859	−0.929058
$592$	0	0
$593$	−7.75159	−0.318320	−0.159160	−	0.987253i	$-0.550879\pi$
$593$	−7.75159	−0.318320	−0.159160	+	0.987253i	$0.550879\pi$
$594$	0	0
$595$	2.32983	0.0955139
$596$	0	0
$597$	−7.40505	−0.303069
$598$	0	0
$599$	6.11584	0.249887	0.124943	−	0.992164i	$-0.460125\pi$
$599$	6.11584	0.249887	0.124943	+	0.992164i	$0.460125\pi$
$600$	0	0
$601$	−5.24488	−0.213943	−0.106972	−	0.994262i	$-0.534115\pi$
$601$	−5.24488	−0.213943	−0.106972	+	0.994262i	$0.534115\pi$
$602$	0	0
$603$	6.23491	0.253905
$604$	0	0
$605$	0.737456	0.0299818
$606$	0	0
$607$	43.2379	1.75497	0.877486	−	0.479602i	$-0.159219\pi$
$607$	43.2379	1.75497	0.877486	+	0.479602i	$0.159219\pi$
$608$	0	0
$609$	1.66456	0.0674514
$610$	0	0
$611$	−1.52688	−0.0617708
$612$	0	0
$613$	−8.73802	−0.352925	−0.176463	−	0.984307i	$-0.556465\pi$
$613$	−8.73802	−0.352925	−0.176463	+	0.984307i	$0.556465\pi$
$614$	0	0
$615$	−0.716042	−0.0288736
$616$	0	0
$617$	−14.1215	−0.568512	−0.284256	−	0.958748i	$-0.591747\pi$
$617$	−14.1215	−0.568512	−0.284256	+	0.958748i	$0.591747\pi$
$618$	0	0
$619$	6.53825	0.262794	0.131397	−	0.991330i	$-0.458054\pi$
$619$	6.53825	0.262794	0.131397	+	0.991330i	$0.458054\pi$
$620$	0	0
$621$	−0.402422	−0.0161486
$622$	0	0
$623$	−34.5624	−1.38471
$624$	0	0
$625$	24.7770	0.991078
$626$	0	0
$627$	1.42509	0.0569126
$628$	0	0
$629$	22.8841	0.912450
$630$	0	0
$631$	27.4405	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$631$	27.4405	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$632$	0	0
$633$	−6.39991	−0.254374
$634$	0	0
$635$	2.14411	0.0850862
$636$	0	0
$637$	−6.72388	−0.266410
$638$	0	0
$639$	−2.61717	−0.103534
$640$	0	0
$641$	14.8424	0.586239	0.293119	−	0.956076i	$-0.405307\pi$
$641$	14.8424	0.586239	0.293119	+	0.956076i	$0.405307\pi$
$642$	0	0
$643$	−12.3110	−0.485500	−0.242750	−	0.970089i	$-0.578049\pi$
$643$	−12.3110	−0.485500	−0.242750	+	0.970089i	$0.578049\pi$
$644$	0	0
$645$	0.301230	0.0118609
$646$	0	0
$647$	9.49024	0.373100	0.186550	−	0.982445i	$-0.440269\pi$
$647$	9.49024	0.373100	0.186550	+	0.982445i	$0.440269\pi$
$648$	0	0
$649$	−0.373383	−0.0146566
$650$	0	0
$651$	25.4983	0.999359
$652$	0	0
$653$	−5.78958	−0.226564	−0.113282	−	0.993563i	$-0.536136\pi$
$653$	−5.78958	−0.226564	−0.113282	+	0.993563i	$0.536136\pi$
$654$	0	0
$655$	2.27299	0.0888132
$656$	0	0
$657$	4.65157	0.181475
$658$	0	0
$659$	−16.7205	−0.651340	−0.325670	−	0.945484i	$-0.605590\pi$
$659$	−16.7205	−0.651340	−0.325670	+	0.945484i	$0.605590\pi$
$660$	0	0
$661$	23.7031	0.921944	0.460972	−	0.887415i	$-0.347501\pi$
$661$	23.7031	0.921944	0.460972	+	0.887415i	$0.347501\pi$
$662$	0	0
$663$	7.33922	0.285032
$664$	0	0
$665$	0.271028	0.0105100
$666$	0	0
$667$	0.193035	0.00747433
$668$	0	0
$669$	2.33822	0.0904007
$670$	0	0
$671$	19.9733	0.771060
$672$	0	0
$673$	−30.9062	−1.19135	−0.595673	−	0.803227i	$-0.703115\pi$
$673$	−30.9062	−1.19135	−0.595673	+	0.803227i	$0.703115\pi$
$674$	0	0
$675$	4.98512	0.191877
$676$	0	0
$677$	−26.8031	−1.03013	−0.515063	−	0.857152i	$-0.672232\pi$
$677$	−26.8031	−1.03013	−0.515063	+	0.857152i	$0.672232\pi$
$678$	0	0
$679$	0.0654190	0.00251055
$680$	0	0
$681$	18.9294	0.725377
$682$	0	0
$683$	−11.2809	−0.431650	−0.215825	−	0.976432i	$-0.569244\pi$
$683$	−11.2809	−0.431650	−0.215825	+	0.976432i	$0.569244\pi$
$684$	0	0
$685$	2.53841	0.0969877
$686$	0	0
$687$	18.8437	0.718933
$688$	0	0
$689$	2.62442	0.0999825
$690$	0	0
$691$	−40.1495	−1.52736	−0.763679	−	0.645596i	$-0.776609\pi$
$691$	−40.1495	−1.52736	−0.763679	+	0.645596i	$0.776609\pi$
$692$	0	0
$693$	7.72470	0.293437
$694$	0	0
$695$	0.684342	0.0259586
$696$	0	0
$697$	−32.2993	−1.22342
$698$	0	0
$699$	−7.00542	−0.264969
$700$	0	0
$701$	−10.5205	−0.397356	−0.198678	−	0.980065i	$-0.563665\pi$
$701$	−10.5205	−0.397356	−0.198678	+	0.980065i	$0.563665\pi$
$702$	0	0
$703$	2.66210	0.100403
$704$	0	0
$705$	0.139680	0.00526064
$706$	0	0
$707$	−5.88769	−0.221430
$708$	0	0
$709$	−18.1932	−0.683259	−0.341629	−	0.939835i	$-0.610979\pi$
$709$	−18.1932	−0.683259	−0.341629	+	0.939835i	$0.610979\pi$
$710$	0	0
$711$	15.8128	0.593027
$712$	0	0
$713$	2.95697	0.110740
$714$	0	0
$715$	−0.362184	−0.0135449
$716$	0	0
$717$	2.78325	0.103942
$718$	0	0
$719$	−35.3820	−1.31952	−0.659762	−	0.751474i	$-0.729343\pi$
$719$	−35.3820	−1.31952	−0.659762	+	0.751474i	$0.729343\pi$
$720$	0	0
$721$	−17.0804	−0.636107
$722$	0	0
$723$	−28.9139	−1.07532
$724$	0	0
$725$	−2.39127	−0.0888096
$726$	0	0
$727$	−43.8944	−1.62795	−0.813976	−	0.580899i	$-0.802701\pi$
$727$	−43.8944	−1.62795	−0.813976	+	0.580899i	$0.802701\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.5880	0.502569
$732$	0	0
$733$	10.6718	0.394173	0.197086	−	0.980386i	$-0.436852\pi$
$733$	10.6718	0.394173	0.197086	+	0.980386i	$0.436852\pi$
$734$	0	0
$735$	0.615105	0.0226885
$736$	0	0
$737$	−13.8792	−0.511248
$738$	0	0
$739$	−46.1584	−1.69797	−0.848983	−	0.528421i	$-0.822784\pi$
$739$	−46.1584	−1.69797	−0.848983	+	0.528421i	$0.822784\pi$
$740$	0	0
$741$	0.853767	0.0313639
$742$	0	0
$743$	−29.7649	−1.09197	−0.545984	−	0.837796i	$-0.683844\pi$
$743$	−29.7649	−1.09197	−0.545984	+	0.837796i	$0.683844\pi$
$744$	0	0
$745$	0.647627	0.0237272
$746$	0	0
$747$	4.00231	0.146437
$748$	0	0
$749$	24.9161	0.910413
$750$	0	0
$751$	−30.9029	−1.12766	−0.563831	−	0.825890i	$-0.690673\pi$
$751$	−30.9029	−1.12766	−0.563831	+	0.825890i	$0.690673\pi$
$752$	0	0
$753$	−27.6992	−1.00941
$754$	0	0
$755$	−1.09439	−0.0398291
$756$	0	0
$757$	−3.03126	−0.110173	−0.0550864	−	0.998482i	$-0.517543\pi$
$757$	−3.03126	−0.110173	−0.0550864	+	0.998482i	$0.517543\pi$
$758$	0	0
$759$	0.895812	0.0325159
$760$	0	0
$761$	0.378758	0.0137300	0.00686498	−	0.999976i	$-0.497815\pi$
$761$	0.378758	0.0137300	0.00686498	+	0.999976i	$0.497815\pi$
$762$	0	0
$763$	−24.5357	−0.888251
$764$	0	0
$765$	−0.671396	−0.0242744
$766$	0	0
$767$	−0.223693	−0.00807708
$768$	0	0
$769$	1.76437	0.0636248	0.0318124	−	0.999494i	$-0.489872\pi$
$769$	1.76437	0.0636248	0.0318124	+	0.999494i	$0.489872\pi$
$770$	0	0
$771$	−7.07503	−0.254801
$772$	0	0
$773$	29.6387	1.06603	0.533015	−	0.846106i	$-0.321059\pi$
$773$	29.6387	1.06603	0.533015	+	0.846106i	$0.321059\pi$
$774$	0	0
$775$	−36.6303	−1.31580
$776$	0	0
$777$	14.4299	0.517670
$778$	0	0
$779$	−3.75737	−0.134622
$780$	0	0
$781$	5.82595	0.208469
$782$	0	0
$783$	−0.479682	−0.0171424
$784$	0	0
$785$	0.745688	0.0266147
$786$	0	0
$787$	17.5054	0.624001	0.312001	−	0.950082i	$-0.399001\pi$
$787$	17.5054	0.624001	0.312001	+	0.950082i	$0.399001\pi$
$788$	0	0
$789$	14.5697	0.518694
$790$	0	0
$791$	−37.3169	−1.32684
$792$	0	0
$793$	11.9659	0.424923
$794$	0	0
$795$	−0.240084	−0.00851490
$796$	0	0
$797$	14.9100	0.528138	0.264069	−	0.964504i	$-0.414935\pi$
$797$	14.9100	0.528138	0.264069	+	0.964504i	$0.414935\pi$
$798$	0	0
$799$	6.30069	0.222902
$800$	0	0
$801$	9.95997	0.351918
$802$	0	0
$803$	−10.3546	−0.365407
$804$	0	0
$805$	0.170369	0.00600470
$806$	0	0
$807$	13.6707	0.481230
$808$	0	0
$809$	52.4836	1.84523	0.922613	−	0.385726i	$-0.126049\pi$
$809$	52.4836	1.84523	0.922613	+	0.385726i	$0.126049\pi$
$810$	0	0
$811$	−20.8771	−0.733095	−0.366547	−	0.930399i	$-0.619460\pi$
$811$	−20.8771	−0.733095	−0.366547	+	0.930399i	$0.619460\pi$
$812$	0	0
$813$	19.4027	0.680483
$814$	0	0
$815$	2.23581	0.0783171
$816$	0	0
$817$	1.58068	0.0553010
$818$	0	0
$819$	4.62784	0.161710
$820$	0	0
$821$	−50.0630	−1.74721	−0.873606	−	0.486634i	$-0.838225\pi$
$821$	−50.0630	−1.74721	−0.873606	+	0.486634i	$0.838225\pi$
$822$	0	0
$823$	2.35960	0.0822504	0.0411252	−	0.999154i	$-0.486906\pi$
$823$	2.35960	0.0822504	0.0411252	+	0.999154i	$0.486906\pi$
$824$	0	0
$825$	−11.0971	−0.386352
$826$	0	0
$827$	30.7779	1.07025	0.535127	−	0.844772i	$-0.320264\pi$
$827$	30.7779	1.07025	0.535127	+	0.844772i	$0.320264\pi$
$828$	0	0
$829$	33.4812	1.16285	0.581425	−	0.813600i	$-0.302496\pi$
$829$	33.4812	1.16285	0.581425	+	0.813600i	$0.302496\pi$
$830$	0	0
$831$	32.8515	1.13961
$832$	0	0
$833$	27.7463	0.961351
$834$	0	0
$835$	−0.122001	−0.00422200
$836$	0	0
$837$	−7.34794	−0.253982
$838$	0	0
$839$	−8.72318	−0.301158	−0.150579	−	0.988598i	$-0.548114\pi$
$839$	−8.72318	−0.301158	−0.150579	+	0.988598i	$0.548114\pi$
$840$	0	0
$841$	−28.7699	−0.992066
$842$	0	0
$843$	24.2094	0.833817
$844$	0	0
$845$	1.36902	0.0470958
$846$	0	0
$847$	20.9759	0.720740
$848$	0	0
$849$	20.9337	0.718442
$850$	0	0
$851$	1.67340	0.0573633
$852$	0	0
$853$	17.1160	0.586042	0.293021	−	0.956106i	$-0.405339\pi$
$853$	17.1160	0.586042	0.293021	+	0.956106i	$0.405339\pi$
$854$	0	0
$855$	−0.0781032	−0.00267107
$856$	0	0
$857$	45.7436	1.56257	0.781285	−	0.624174i	$-0.214564\pi$
$857$	45.7436	1.56257	0.781285	+	0.624174i	$0.214564\pi$
$858$	0	0
$859$	−38.3993	−1.31017	−0.655083	−	0.755557i	$-0.727366\pi$
$859$	−38.3993	−1.31017	−0.655083	+	0.755557i	$0.727366\pi$
$860$	0	0
$861$	−20.3668	−0.694098
$862$	0	0
$863$	42.0073	1.42994	0.714972	−	0.699153i	$-0.246439\pi$
$863$	42.0073	1.42994	0.714972	+	0.699153i	$0.246439\pi$
$864$	0	0
$865$	−1.03367	−0.0351458
$866$	0	0
$867$	−13.2855	−0.451198
$868$	0	0
$869$	−35.2001	−1.19408
$870$	0	0
$871$	−8.31502	−0.281744
$872$	0	0
$873$	−0.0188520	−0.000638044 0
$874$	0	0
$875$	−4.22728	−0.142908
$876$	0	0
$877$	32.2213	1.08804	0.544018	−	0.839074i	$-0.316902\pi$
$877$	32.2213	1.08804	0.544018	+	0.839074i	$0.316902\pi$
$878$	0	0
$879$	−28.6411	−0.966040
$880$	0	0
$881$	−48.8639	−1.64627	−0.823134	−	0.567847i	$-0.807777\pi$
$881$	−48.8639	−1.64627	−0.823134	+	0.567847i	$0.807777\pi$
$882$	0	0
$883$	4.78312	0.160965	0.0804823	−	0.996756i	$-0.474354\pi$
$883$	4.78312	0.160965	0.0804823	+	0.996756i	$0.474354\pi$
$884$	0	0
$885$	0.0204636	0.000687875 0
$886$	0	0
$887$	40.9773	1.37588	0.687942	−	0.725766i	$-0.258514\pi$
$887$	40.9773	1.37588	0.687942	+	0.725766i	$0.258514\pi$
$888$	0	0
$889$	60.9860	2.04541
$890$	0	0
$891$	−2.22605	−0.0745755
$892$	0	0
$893$	0.732956	0.0245274
$894$	0	0
$895$	2.31210	0.0772850
$896$	0	0
$897$	0.536679	0.0179192
$898$	0	0
$899$	3.52468	0.117555
$900$	0	0
$901$	−10.8297	−0.360791
$902$	0	0
$903$	8.56807	0.285128
$904$	0	0
$905$	−1.86443	−0.0619759
$906$	0	0
$907$	−58.3280	−1.93675	−0.968374	−	0.249503i	$-0.919733\pi$
$907$	−58.3280	−1.93675	−0.968374	+	0.249503i	$0.919733\pi$
$908$	0	0
$909$	1.69668	0.0562752
$910$	0	0
$911$	−14.7622	−0.489092	−0.244546	−	0.969638i	$-0.578639\pi$
$911$	−14.7622	−0.489092	−0.244546	+	0.969638i	$0.578639\pi$
$912$	0	0
$913$	−8.90935	−0.294856
$914$	0	0
$915$	−1.09465	−0.0361881
$916$	0	0
$917$	64.6521	2.13500
$918$	0	0
$919$	−14.9191	−0.492135	−0.246067	−	0.969253i	$-0.579138\pi$
$919$	−14.9191	−0.492135	−0.246067	+	0.969253i	$0.579138\pi$
$920$	0	0
$921$	21.0765	0.694494
$922$	0	0
$923$	3.49031	0.114885
$924$	0	0
$925$	−20.7297	−0.681587
$926$	0	0
$927$	4.92211	0.161663
$928$	0	0
$929$	32.9389	1.08069	0.540345	−	0.841443i	$-0.318294\pi$
$929$	32.9389	1.08069	0.540345	+	0.841443i	$0.318294\pi$
$930$	0	0
$931$	3.22771	0.105784
$932$	0	0
$933$	29.8062	0.975810
$934$	0	0
$935$	1.49456	0.0488774
$936$	0	0
$937$	−11.7291	−0.383175	−0.191587	−	0.981476i	$-0.561363\pi$
$937$	−11.7291	−0.383175	−0.191587	+	0.981476i	$0.561363\pi$
$938$	0	0
$939$	16.9090	0.551803
$940$	0	0
$941$	−8.15429	−0.265822	−0.132911	−	0.991128i	$-0.542432\pi$
$941$	−8.15429	−0.265822	−0.132911	+	0.991128i	$0.542432\pi$
$942$	0	0
$943$	−2.36188	−0.0769134
$944$	0	0
$945$	−0.423358	−0.0137718
$946$	0	0
$947$	−6.86175	−0.222977	−0.111488	−	0.993766i	$-0.535562\pi$
$947$	−6.86175	−0.222977	−0.111488	+	0.993766i	$0.535562\pi$
$948$	0	0
$949$	−6.20343	−0.201372
$950$	0	0
$951$	4.23258	0.137251
$952$	0	0
$953$	51.6273	1.67237	0.836186	−	0.548446i	$-0.184780\pi$
$953$	51.6273	1.67237	0.836186	+	0.548446i	$0.184780\pi$
$954$	0	0
$955$	−0.501974	−0.0162435
$956$	0	0
$957$	1.06780	0.0345170
$958$	0	0
$959$	72.2015	2.33151
$960$	0	0
$961$	22.9923	0.741686
$962$	0	0
$963$	−7.18014	−0.231377
$964$	0	0
$965$	1.55809	0.0501565
$966$	0	0
$967$	11.9545	0.384430	0.192215	−	0.981353i	$-0.438433\pi$
$967$	11.9545	0.384430	0.192215	+	0.981353i	$0.438433\pi$
$968$	0	0
$969$	−3.52309	−0.113178
$970$	0	0
$971$	−26.6958	−0.856709	−0.428355	−	0.903611i	$-0.640907\pi$
$971$	−26.6958	−0.856709	−0.428355	+	0.903611i	$0.640907\pi$
$972$	0	0
$973$	19.4651	0.624024
$974$	0	0
$975$	−6.64826	−0.212915
$976$	0	0
$977$	23.5330	0.752886	0.376443	−	0.926440i	$-0.377147\pi$
$977$	23.5330	0.752886	0.376443	+	0.926440i	$0.377147\pi$
$978$	0	0
$979$	−22.1714	−0.708601
$980$	0	0
$981$	7.07053	0.225745
$982$	0	0
$983$	50.4386	1.60874	0.804371	−	0.594127i	$-0.202502\pi$
$983$	50.4386	1.60874	0.804371	+	0.594127i	$0.202502\pi$
$984$	0	0
$985$	−2.75549	−0.0877971
$986$	0	0
$987$	3.97299	0.126462
$988$	0	0
$989$	0.993616	0.0315952
$990$	0	0
$991$	48.2954	1.53415	0.767077	−	0.641555i	$-0.221710\pi$
$991$	48.2954	1.53415	0.767077	+	0.641555i	$0.221710\pi$
$992$	0	0
$993$	14.9764	0.475263
$994$	0	0
$995$	−0.903421	−0.0286404
$996$	0	0
$997$	−11.8212	−0.374381	−0.187191	−	0.982324i	$-0.559938\pi$
$997$	−11.8212	−0.374381	−0.187191	+	0.982324i	$0.559938\pi$
$998$	0	0
$999$	−4.15831	−0.131563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.x.1.3		8
4.3	odd	2		501.2.a.e.1.6	✓	8
12.11	even	2		1503.2.a.e.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.e.1.6	✓	8	4.3	odd	2
1503.2.a.e.1.3		8	12.11	even	2
8016.2.a.x.1.3		8	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.122001 q^{5} -3.47013 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.122001 q^{5} -3.47013 q^{7} +1.00000 q^{9} -2.22605 q^{11} -1.33362 q^{13} +0.122001 q^{15} +5.50322 q^{17} +0.640187 q^{19} +3.47013 q^{21} +0.402422 q^{23} -4.98512 q^{25} -1.00000 q^{27} +0.479682 q^{29} +7.34794 q^{31} +2.22605 q^{33} +0.423358 q^{35} +4.15831 q^{37} +1.33362 q^{39} -5.86917 q^{41} +2.46909 q^{43} -0.122001 q^{45} +1.14491 q^{47} +5.04182 q^{49} -5.50322 q^{51} -1.96789 q^{53} +0.271580 q^{55} -0.640187 q^{57} +0.167733 q^{59} -8.97251 q^{61} -3.47013 q^{63} +0.162703 q^{65} +6.23491 q^{67} -0.402422 q^{69} -2.61717 q^{71} +4.65157 q^{73} +4.98512 q^{75} +7.72470 q^{77} +15.8128 q^{79} +1.00000 q^{81} +4.00231 q^{83} -0.671396 q^{85} -0.479682 q^{87} +9.95997 q^{89} +4.62784 q^{91} -7.34794 q^{93} -0.0781032 q^{95} -0.0188520 q^{97} -2.22605 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9	\( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9 - 13 * q^11 - 7 * q^15 + 11 * q^17 - 12 * q^19 - 4 * q^21 - 7 * q^23 - 5 * q^25 - 8 * q^27 + q^29 + 2 * q^31 + 13 * q^33 + 4 * q^35 - 9 * q^37 + 4 * q^41 - 2 * q^43 + 7 * q^45 - 17 * q^47 - 2 * q^49 - 11 * q^51 + 9 * q^53 - 7 * q^55 + 12 * q^57 - 29 * q^59 - 12 * q^61 + 4 * q^63 + 8 * q^65 + 7 * q^69 - 13 * q^71 - 20 * q^73 + 5 * q^75 - 22 * q^77 - 8 * q^79 + 8 * q^81 - 33 * q^83 - 31 * q^85 - q^87 + 4 * q^89 - q^91 - 2 * q^93 - 3 * q^95 - 31 * q^97 - 13 * q^99

Introduction

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