LMFDB - Embedded newform 3864.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,2,Mod(1,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3864.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:	$30.8541953410$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.39605.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 9x - 2 $ x^4 - 2x^3 - 8x^2 + 9*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.57685$ of defining polynomial
Character	$\chi$	$=$	3864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.57685 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.57685 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.81840 q^{11} -3.91214 q^{13} -3.57685 q^{15} -7.21699 q^{17} -0.335294 q^{19} -1.00000 q^{21} -1.00000 q^{23} +7.79384 q^{25} +1.00000 q^{27} +7.11144 q^{29} -1.21699 q^{31} -4.81840 q^{33} +3.57685 q^{35} +1.55915 q^{37} -3.91214 q^{39} -0.0422591 q^{41} +8.12325 q^{43} -3.57685 q^{45} +4.10555 q^{47} +1.00000 q^{49} -7.21699 q^{51} +6.35299 q^{53} +17.2347 q^{55} -0.335294 q^{57} -2.08099 q^{59} +10.9603 q^{61} -1.00000 q^{63} +13.9931 q^{65} +1.19929 q^{67} -1.00000 q^{69} -6.89939 q^{71} -2.73388 q^{73} +7.79384 q^{75} +4.81840 q^{77} +16.5582 q^{79} +1.00000 q^{81} -13.0413 q^{83} +25.8141 q^{85} +7.11144 q^{87} -9.17139 q^{89} +3.91214 q^{91} -1.21699 q^{93} +1.19929 q^{95} -13.7820 q^{97} -4.81840 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{21} - 4 q^{23} + 4 q^{27} + 11 q^{29} + 14 q^{31} + 2 q^{35} + 13 q^{37} + 2 q^{39} + 7 q^{41} + 12 q^{43} - 2 q^{45} + 15 q^{47} + 4 q^{49} - 10 q^{51} + q^{53} + 31 q^{55} + 4 q^{57} + 5 q^{59} + 3 q^{61} - 4 q^{63} + 25 q^{65} + 5 q^{67} - 4 q^{69} + 5 q^{71} - 6 q^{73} + 26 q^{79} + 4 q^{81} + 2 q^{83} + 5 q^{85} + 11 q^{87} + 7 q^{89} - 2 q^{91} + 14 q^{93} + 5 q^{95} - 27 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^13 - 2 * q^15 - 10 * q^17 + 4 * q^19 - 4 * q^21 - 4 * q^23 + 4 * q^27 + 11 * q^29 + 14 * q^31 + 2 * q^35 + 13 * q^37 + 2 * q^39 + 7 * q^41 + 12 * q^43 - 2 * q^45 + 15 * q^47 + 4 * q^49 - 10 * q^51 + q^53 + 31 * q^55 + 4 * q^57 + 5 * q^59 + 3 * q^61 - 4 * q^63 + 25 * q^65 + 5 * q^67 - 4 * q^69 + 5 * q^71 - 6 * q^73 + 26 * q^79 + 4 * q^81 + 2 * q^83 + 5 * q^85 + 11 * q^87 + 7 * q^89 - 2 * q^91 + 14 * q^93 + 5 * q^95 - 27 * q^97

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
$5$	−3.57685	−1.59961	−0.799807	−	0.600257i	$-0.795065\pi$
$5$	−3.57685	−1.59961	−0.799807	+	0.600257i	$0.795065\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.81840	−1.45280	−0.726401	−	0.687271i	$-0.758809\pi$
$11$	−4.81840	−1.45280	−0.726401	+	0.687271i	$0.758809\pi$
$12$	0	0
$13$	−3.91214	−1.08503	−0.542516	−	0.840045i	$-0.682528\pi$
$13$	−3.91214	−1.08503	−0.542516	+	0.840045i	$0.682528\pi$
$14$	0	0
$15$	−3.57685	−0.923538
$16$	0	0
$17$	−7.21699	−1.75038	−0.875189	−	0.483782i	$-0.839263\pi$
$17$	−7.21699	−1.75038	−0.875189	+	0.483782i	$0.839263\pi$
$18$	0	0
$19$	−0.335294	−0.0769217	−0.0384608	−	0.999260i	$-0.512245\pi$
$19$	−0.335294	−0.0769217	−0.0384608	+	0.999260i	$0.512245\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	7.79384	1.55877
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.11144	1.32056	0.660280	−	0.751019i	$-0.270438\pi$
$29$	7.11144	1.32056	0.660280	+	0.751019i	$0.270438\pi$
$30$	0	0
$31$	−1.21699	−0.218578	−0.109289	−	0.994010i	$-0.534857\pi$
$31$	−1.21699	−0.218578	−0.109289	+	0.994010i	$0.534857\pi$
$32$	0	0
$33$	−4.81840	−0.838776
$34$	0	0
$35$	3.57685	0.604598
$36$	0	0
$37$	1.55915	0.256323	0.128161	−	0.991753i	$-0.459092\pi$
$37$	1.55915	0.256323	0.128161	+	0.991753i	$0.459092\pi$
$38$	0	0
$39$	−3.91214	−0.626444
$40$	0	0
$41$	−0.0422591	−0.00659976	−0.00329988	−	0.999995i	$-0.501050\pi$
$41$	−0.0422591	−0.00659976	−0.00329988	+	0.999995i	$0.501050\pi$
$42$	0	0
$43$	8.12325	1.23878	0.619392	−	0.785082i	$-0.287379\pi$
$43$	8.12325	1.23878	0.619392	+	0.785082i	$0.287379\pi$
$44$	0	0
$45$	−3.57685	−0.533205
$46$	0	0
$47$	4.10555	0.598857	0.299428	−	0.954119i	$-0.403204\pi$
$47$	4.10555	0.598857	0.299428	+	0.954119i	$0.403204\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.21699	−1.01058
$52$	0	0
$53$	6.35299	0.872650	0.436325	−	0.899789i	$-0.356280\pi$
$53$	6.35299	0.872650	0.436325	+	0.899789i	$0.356280\pi$
$54$	0	0
$55$	17.2347	2.32392
$56$	0	0
$57$	−0.335294	−0.0444107
$58$	0	0
$59$	−2.08099	−0.270922	−0.135461	−	0.990783i	$-0.543252\pi$
$59$	−2.08099	−0.270922	−0.135461	+	0.990783i	$0.543252\pi$
$60$	0	0
$61$	10.9603	1.40332	0.701660	−	0.712512i	$-0.252443\pi$
$61$	10.9603	1.40332	0.701660	+	0.712512i	$0.252443\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	13.9931	1.73563
$66$	0	0
$67$	1.19929	0.146517	0.0732586	−	0.997313i	$-0.476660\pi$
$67$	1.19929	0.146517	0.0732586	+	0.997313i	$0.476660\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.89939	−0.818807	−0.409404	−	0.912353i	$-0.634263\pi$
$71$	−6.89939	−0.818807	−0.409404	+	0.912353i	$0.634263\pi$
$72$	0	0
$73$	−2.73388	−0.319977	−0.159988	−	0.987119i	$-0.551146\pi$
$73$	−2.73388	−0.319977	−0.159988	+	0.987119i	$0.551146\pi$
$74$	0	0
$75$	7.79384	0.899955
$76$	0	0
$77$	4.81840	0.549108
$78$	0	0
$79$	16.5582	1.86294	0.931470	−	0.363819i	$-0.118527\pi$
$79$	16.5582	1.86294	0.931470	+	0.363819i	$0.118527\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.0413	−1.43147	−0.715733	−	0.698374i	$-0.753907\pi$
$83$	−13.0413	−1.43147	−0.715733	+	0.698374i	$0.753907\pi$
$84$	0	0
$85$	25.8141	2.79993
$86$	0	0
$87$	7.11144	0.762426
$88$	0	0
$89$	−9.17139	−0.972165	−0.486083	−	0.873913i	$-0.661575\pi$
$89$	−9.17139	−0.972165	−0.486083	+	0.873913i	$0.661575\pi$
$90$	0	0
$91$	3.91214	0.410104
$92$	0	0
$93$	−1.21699	−0.126196
$94$	0	0
$95$	1.19929	0.123045
$96$	0	0
$97$	−13.7820	−1.39935	−0.699676	−	0.714460i	$-0.746672\pi$
$97$	−13.7820	−1.39935	−0.699676	+	0.714460i	$0.746672\pi$
$98$	0	0
$99$	−4.81840	−0.484268
$100$	0	0
$101$	15.7178	1.56398	0.781989	−	0.623292i	$-0.214205\pi$
$101$	15.7178	1.56398	0.781989	+	0.623292i	$0.214205\pi$
$102$	0	0
$103$	3.70697	0.365258	0.182629	−	0.983182i	$-0.441539\pi$
$103$	3.70697	0.365258	0.182629	+	0.983182i	$0.441539\pi$
$104$	0	0
$105$	3.57685	0.349065
$106$	0	0
$107$	−2.65289	−0.256465	−0.128232	−	0.991744i	$-0.540930\pi$
$107$	−2.65289	−0.256465	−0.128232	+	0.991744i	$0.540930\pi$
$108$	0	0
$109$	−6.85713	−0.656794	−0.328397	−	0.944540i	$-0.606508\pi$
$109$	−6.85713	−0.656794	−0.328397	+	0.944540i	$0.606508\pi$
$110$	0	0
$111$	1.55915	0.147988
$112$	0	0
$113$	8.54306	0.803664	0.401832	−	0.915713i	$-0.368374\pi$
$113$	8.54306	0.803664	0.401832	+	0.915713i	$0.368374\pi$
$114$	0	0
$115$	3.57685	0.333543
$116$	0	0
$117$	−3.91214	−0.361678
$118$	0	0
$119$	7.21699	0.661580
$120$	0	0
$121$	12.2170	1.11064
$122$	0	0
$123$	−0.0422591	−0.00381037
$124$	0	0
$125$	−9.99313	−0.893813
$126$	0	0
$127$	−15.7718	−1.39952	−0.699761	−	0.714377i	$-0.746710\pi$
$127$	−15.7718	−1.39952	−0.699761	+	0.714377i	$0.746710\pi$
$128$	0	0
$129$	8.12325	0.715212
$130$	0	0
$131$	−12.0987	−1.05707	−0.528534	−	0.848912i	$-0.677258\pi$
$131$	−12.0987	−1.05707	−0.528534	+	0.848912i	$0.677258\pi$
$132$	0	0
$133$	0.335294	0.0290737
$134$	0	0
$135$	−3.57685	−0.307846
$136$	0	0
$137$	−21.7145	−1.85519	−0.927595	−	0.373586i	$-0.878128\pi$
$137$	−21.7145	−1.85519	−0.927595	+	0.373586i	$0.878128\pi$
$138$	0	0
$139$	7.30485	0.619589	0.309795	−	0.950804i	$-0.399740\pi$
$139$	7.30485	0.619589	0.309795	+	0.950804i	$0.399740\pi$
$140$	0	0
$141$	4.10555	0.345750
$142$	0	0
$143$	18.8503	1.57634
$144$	0	0
$145$	−25.4365	−2.11239
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	3.39030	0.277744	0.138872	−	0.990310i	$-0.455652\pi$
$149$	3.39030	0.277744	0.138872	+	0.990310i	$0.455652\pi$
$150$	0	0
$151$	−18.8469	−1.53374	−0.766871	−	0.641802i	$-0.778187\pi$
$151$	−18.8469	−1.53374	−0.766871	+	0.641802i	$0.778187\pi$
$152$	0	0
$153$	−7.21699	−0.583459
$154$	0	0
$155$	4.35299	0.349641
$156$	0	0
$157$	13.8243	1.10330	0.551649	−	0.834076i	$-0.313999\pi$
$157$	13.8243	1.10330	0.551649	+	0.834076i	$0.313999\pi$
$158$	0	0
$159$	6.35299	0.503825
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	13.7433	1.07646	0.538229	−	0.842799i	$-0.319094\pi$
$163$	13.7433	1.07646	0.538229	+	0.842799i	$0.319094\pi$
$164$	0	0
$165$	17.2347	1.34172
$166$	0	0
$167$	−0.335294	−0.0259458	−0.0129729	−	0.999916i	$-0.504130\pi$
$167$	−0.335294	−0.0259458	−0.0129729	+	0.999916i	$0.504130\pi$
$168$	0	0
$169$	2.30485	0.177296
$170$	0	0
$171$	−0.335294	−0.0256406
$172$	0	0
$173$	0.323531	0.0245976	0.0122988	−	0.999924i	$-0.496085\pi$
$173$	0.323531	0.0245976	0.0122988	+	0.999924i	$0.496085\pi$
$174$	0	0
$175$	−7.79384	−0.589159
$176$	0	0
$177$	−2.08099	−0.156417
$178$	0	0
$179$	5.06824	0.378818	0.189409	−	0.981898i	$-0.439343\pi$
$179$	5.06824	0.378818	0.189409	+	0.981898i	$0.439343\pi$
$180$	0	0
$181$	−12.5818	−0.935197	−0.467599	−	0.883941i	$-0.654881\pi$
$181$	−12.5818	−0.935197	−0.467599	+	0.883941i	$0.654881\pi$
$182$	0	0
$183$	10.9603	0.810207
$184$	0	0
$185$	−5.57685	−0.410018
$186$	0	0
$187$	34.7744	2.54295
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	7.06918	0.511508	0.255754	−	0.966742i	$-0.417676\pi$
$191$	7.06918	0.511508	0.255754	+	0.966742i	$0.417676\pi$
$192$	0	0
$193$	20.4247	1.47020	0.735101	−	0.677957i	$-0.237135\pi$
$193$	20.4247	1.47020	0.735101	+	0.677957i	$0.237135\pi$
$194$	0	0
$195$	13.9931	1.00207
$196$	0	0
$197$	−6.87836	−0.490063	−0.245031	−	0.969515i	$-0.578798\pi$
$197$	−6.87836	−0.490063	−0.245031	+	0.969515i	$0.578798\pi$
$198$	0	0
$199$	−18.3182	−1.29854	−0.649272	−	0.760556i	$-0.724926\pi$
$199$	−18.3182	−1.29854	−0.649272	+	0.760556i	$0.724926\pi$
$200$	0	0
$201$	1.19929	0.0845917
$202$	0	0
$203$	−7.11144	−0.499125
$204$	0	0
$205$	0.151154	0.0105571
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	1.61558	0.111752
$210$	0	0
$211$	18.1596	1.25016	0.625078	−	0.780562i	$-0.285067\pi$
$211$	18.1596	1.25016	0.625078	+	0.780562i	$0.285067\pi$
$212$	0	0
$213$	−6.89939	−0.472739
$214$	0	0
$215$	−29.0556	−1.98158
$216$	0	0
$217$	1.21699	0.0826147
$218$	0	0
$219$	−2.73388	−0.184739
$220$	0	0
$221$	28.2339	1.89922
$222$	0	0
$223$	25.5421	1.71042	0.855212	−	0.518278i	$-0.173427\pi$
$223$	25.5421	1.71042	0.855212	+	0.518278i	$0.173427\pi$
$224$	0	0
$225$	7.79384	0.519589
$226$	0	0
$227$	16.7423	1.11123	0.555613	−	0.831441i	$-0.312484\pi$
$227$	16.7423	1.11123	0.555613	+	0.831441i	$0.312484\pi$
$228$	0	0
$229$	26.5504	1.75450	0.877249	−	0.480036i	$-0.159376\pi$
$229$	26.5504	1.75450	0.877249	+	0.480036i	$0.159376\pi$
$230$	0	0
$231$	4.81840	0.317028
$232$	0	0
$233$	5.76692	0.377804	0.188902	−	0.981996i	$-0.439507\pi$
$233$	5.76692	0.377804	0.188902	+	0.981996i	$0.439507\pi$
$234$	0	0
$235$	−14.6849	−0.957940
$236$	0	0
$237$	16.5582	1.07557
$238$	0	0
$239$	23.8023	1.53964	0.769822	−	0.638259i	$-0.220345\pi$
$239$	23.8023	1.53964	0.769822	+	0.638259i	$0.220345\pi$
$240$	0	0
$241$	−12.1031	−0.779632	−0.389816	−	0.920893i	$-0.627461\pi$
$241$	−12.1031	−0.779632	−0.389816	+	0.920893i	$0.627461\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.57685	−0.228516
$246$	0	0
$247$	1.31172	0.0834625
$248$	0	0
$249$	−13.0413	−0.826457
$250$	0	0
$251$	−7.80993	−0.492958	−0.246479	−	0.969148i	$-0.579274\pi$
$251$	−7.80993	−0.492958	−0.246479	+	0.969148i	$0.579274\pi$
$252$	0	0
$253$	4.81840	0.302930
$254$	0	0
$255$	25.8141	1.61654
$256$	0	0
$257$	−2.26024	−0.140990	−0.0704948	−	0.997512i	$-0.522458\pi$
$257$	−2.26024	−0.140990	−0.0704948	+	0.997512i	$0.522458\pi$
$258$	0	0
$259$	−1.55915	−0.0968810
$260$	0	0
$261$	7.11144	0.440187
$262$	0	0
$263$	−20.0168	−1.23429	−0.617143	−	0.786851i	$-0.711710\pi$
$263$	−20.0168	−1.23429	−0.617143	+	0.786851i	$0.711710\pi$
$264$	0	0
$265$	−22.7237	−1.39590
$266$	0	0
$267$	−9.17139	−0.561280
$268$	0	0
$269$	30.1730	1.83968	0.919840	−	0.392294i	$-0.128318\pi$
$269$	30.1730	1.83968	0.919840	+	0.392294i	$0.128318\pi$
$270$	0	0
$271$	19.5735	1.18901	0.594503	−	0.804093i	$-0.297349\pi$
$271$	19.5735	1.18901	0.594503	+	0.804093i	$0.297349\pi$
$272$	0	0
$273$	3.91214	0.236774
$274$	0	0
$275$	−37.5538	−2.26458
$276$	0	0
$277$	−22.1140	−1.32870	−0.664350	−	0.747422i	$-0.731292\pi$
$277$	−22.1140	−1.32870	−0.664350	+	0.747422i	$0.731292\pi$
$278$	0	0
$279$	−1.21699	−0.0728593
$280$	0	0
$281$	−21.7424	−1.29704	−0.648520	−	0.761197i	$-0.724612\pi$
$281$	−21.7424	−1.29704	−0.648520	+	0.761197i	$0.724612\pi$
$282$	0	0
$283$	5.59788	0.332760	0.166380	−	0.986062i	$-0.446792\pi$
$283$	5.59788	0.332760	0.166380	+	0.986062i	$0.446792\pi$
$284$	0	0
$285$	1.19929	0.0710401
$286$	0	0
$287$	0.0422591	0.00249447
$288$	0	0
$289$	35.0850	2.06382
$290$	0	0
$291$	−13.7820	−0.807917
$292$	0	0
$293$	−14.6706	−0.857065	−0.428532	−	0.903526i	$-0.640969\pi$
$293$	−14.6706	−0.857065	−0.428532	+	0.903526i	$0.640969\pi$
$294$	0	0
$295$	7.44339	0.433371
$296$	0	0
$297$	−4.81840	−0.279592
$298$	0	0
$299$	3.91214	0.226245
$300$	0	0
$301$	−8.12325	−0.468216
$302$	0	0
$303$	15.7178	0.902964
$304$	0	0
$305$	−39.2033	−2.24477
$306$	0	0
$307$	9.07604	0.517997	0.258999	−	0.965878i	$-0.416607\pi$
$307$	9.07604	0.517997	0.258999	+	0.965878i	$0.416607\pi$
$308$	0	0
$309$	3.70697	0.210882
$310$	0	0
$311$	−4.10221	−0.232615	−0.116308	−	0.993213i	$-0.537106\pi$
$311$	−4.10221	−0.232615	−0.116308	+	0.993213i	$0.537106\pi$
$312$	0	0
$313$	2.09628	0.118489	0.0592444	−	0.998244i	$-0.481131\pi$
$313$	2.09628	0.118489	0.0592444	+	0.998244i	$0.481131\pi$
$314$	0	0
$315$	3.57685	0.201533
$316$	0	0
$317$	10.9063	0.612557	0.306278	−	0.951942i	$-0.400916\pi$
$317$	10.9063	0.612557	0.306278	+	0.951942i	$0.400916\pi$
$318$	0	0
$319$	−34.2658	−1.91851
$320$	0	0
$321$	−2.65289	−0.148070
$322$	0	0
$323$	2.41981	0.134642
$324$	0	0
$325$	−30.4906	−1.69131
$326$	0	0
$327$	−6.85713	−0.379200
$328$	0	0
$329$	−4.10555	−0.226347
$330$	0	0
$331$	29.6368	1.62899	0.814493	−	0.580173i	$-0.197015\pi$
$331$	29.6368	1.62899	0.814493	+	0.580173i	$0.197015\pi$
$332$	0	0
$333$	1.55915	0.0854410
$334$	0	0
$335$	−4.28969	−0.234371
$336$	0	0
$337$	−31.0731	−1.69266	−0.846331	−	0.532658i	$-0.821193\pi$
$337$	−31.0731	−1.69266	−0.846331	+	0.532658i	$0.821193\pi$
$338$	0	0
$339$	8.54306	0.463995
$340$	0	0
$341$	5.86395	0.317551
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.57685	0.192571
$346$	0	0
$347$	13.9019	0.746295	0.373147	−	0.927772i	$-0.378279\pi$
$347$	13.9019	0.746295	0.373147	+	0.927772i	$0.378279\pi$
$348$	0	0
$349$	−15.8868	−0.850403	−0.425201	−	0.905099i	$-0.639797\pi$
$349$	−15.8868	−0.850403	−0.425201	+	0.905099i	$0.639797\pi$
$350$	0	0
$351$	−3.91214	−0.208815
$352$	0	0
$353$	−26.3917	−1.40469	−0.702345	−	0.711837i	$-0.747864\pi$
$353$	−26.3917	−1.40469	−0.702345	+	0.711837i	$0.747864\pi$
$354$	0	0
$355$	24.6781	1.30978
$356$	0	0
$357$	7.21699	0.381964
$358$	0	0
$359$	22.1208	1.16749	0.583747	−	0.811936i	$-0.301586\pi$
$359$	22.1208	1.16749	0.583747	+	0.811936i	$0.301586\pi$
$360$	0	0
$361$	−18.8876	−0.994083
$362$	0	0
$363$	12.2170	0.641226
$364$	0	0
$365$	9.77868	0.511840
$366$	0	0
$367$	2.69756	0.140811	0.0704057	−	0.997518i	$-0.477571\pi$
$367$	2.69756	0.140811	0.0704057	+	0.997518i	$0.477571\pi$
$368$	0	0
$369$	−0.0422591	−0.00219992
$370$	0	0
$371$	−6.35299	−0.329831
$372$	0	0
$373$	−12.2862	−0.636154	−0.318077	−	0.948065i	$-0.603037\pi$
$373$	−12.2862	−0.636154	−0.318077	+	0.948065i	$0.603037\pi$
$374$	0	0
$375$	−9.99313	−0.516043
$376$	0	0
$377$	−27.8209	−1.43285
$378$	0	0
$379$	19.8198	1.01808	0.509038	−	0.860744i	$-0.330001\pi$
$379$	19.8198	1.01808	0.509038	+	0.860744i	$0.330001\pi$
$380$	0	0
$381$	−15.7718	−0.808015
$382$	0	0
$383$	7.92297	0.404845	0.202422	−	0.979298i	$-0.435119\pi$
$383$	7.92297	0.404845	0.202422	+	0.979298i	$0.435119\pi$
$384$	0	0
$385$	−17.2347	−0.878361
$386$	0	0
$387$	8.12325	0.412928
$388$	0	0
$389$	−12.2508	−0.621139	−0.310569	−	0.950551i	$-0.600520\pi$
$389$	−12.2508	−0.621139	−0.310569	+	0.950551i	$0.600520\pi$
$390$	0	0
$391$	7.21699	0.364979
$392$	0	0
$393$	−12.0987	−0.610298
$394$	0	0
$395$	−59.2260	−2.97999
$396$	0	0
$397$	−8.19924	−0.411508	−0.205754	−	0.978604i	$-0.565965\pi$
$397$	−8.19924	−0.411508	−0.205754	+	0.978604i	$0.565965\pi$
$398$	0	0
$399$	0.335294	0.0167857
$400$	0	0
$401$	−5.88758	−0.294012	−0.147006	−	0.989136i	$-0.546964\pi$
$401$	−5.88758	−0.294012	−0.147006	+	0.989136i	$0.546964\pi$
$402$	0	0
$403$	4.76104	0.237164
$404$	0	0
$405$	−3.57685	−0.177735
$406$	0	0
$407$	−7.51262	−0.372387
$408$	0	0
$409$	4.83684	0.239167	0.119583	−	0.992824i	$-0.461844\pi$
$409$	4.83684	0.239167	0.119583	+	0.992824i	$0.461844\pi$
$410$	0	0
$411$	−21.7145	−1.07109
$412$	0	0
$413$	2.08099	0.102399
$414$	0	0
$415$	46.6466	2.28979
$416$	0	0
$417$	7.30485	0.357720
$418$	0	0
$419$	−3.31920	−0.162154	−0.0810769	−	0.996708i	$-0.525836\pi$
$419$	−3.31920	−0.162154	−0.0810769	+	0.996708i	$0.525836\pi$
$420$	0	0
$421$	−29.8209	−1.45338	−0.726692	−	0.686964i	$-0.758943\pi$
$421$	−29.8209	−1.45338	−0.726692	+	0.686964i	$0.758943\pi$
$422$	0	0
$423$	4.10555	0.199619
$424$	0	0
$425$	−56.2481	−2.72843
$426$	0	0
$427$	−10.9603	−0.530405
$428$	0	0
$429$	18.8503	0.910099
$430$	0	0
$431$	8.23611	0.396719	0.198360	−	0.980129i	$-0.436439\pi$
$431$	8.23611	0.396719	0.198360	+	0.980129i	$0.436439\pi$
$432$	0	0
$433$	−17.2101	−0.827066	−0.413533	−	0.910489i	$-0.635705\pi$
$433$	−17.2101	−0.827066	−0.413533	+	0.910489i	$0.635705\pi$
$434$	0	0
$435$	−25.4365	−1.21959
$436$	0	0
$437$	0.335294	0.0160393
$438$	0	0
$439$	23.8200	1.13687	0.568433	−	0.822729i	$-0.307550\pi$
$439$	23.8200	1.13687	0.568433	+	0.822729i	$0.307550\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	37.7050	1.79142	0.895709	−	0.444640i	$-0.146669\pi$
$443$	37.7050	1.79142	0.895709	+	0.444640i	$0.146669\pi$
$444$	0	0
$445$	32.8047	1.55509
$446$	0	0
$447$	3.39030	0.160356
$448$	0	0
$449$	33.1552	1.56469	0.782346	−	0.622843i	$-0.214023\pi$
$449$	33.1552	1.56469	0.782346	+	0.622843i	$0.214023\pi$
$450$	0	0
$451$	0.203621	0.00958815
$452$	0	0
$453$	−18.8469	−0.885506
$454$	0	0
$455$	−13.9931	−0.656008
$456$	0	0
$457$	−8.92062	−0.417289	−0.208644	−	0.977992i	$-0.566905\pi$
$457$	−8.92062	−0.417289	−0.208644	+	0.977992i	$0.566905\pi$
$458$	0	0
$459$	−7.21699	−0.336860
$460$	0	0
$461$	−13.2484	−0.617040	−0.308520	−	0.951218i	$-0.599834\pi$
$461$	−13.2484	−0.617040	−0.308520	+	0.951218i	$0.599834\pi$
$462$	0	0
$463$	12.9357	0.601174	0.300587	−	0.953754i	$-0.402817\pi$
$463$	12.9357	0.601174	0.300587	+	0.953754i	$0.402817\pi$
$464$	0	0
$465$	4.35299	0.201865
$466$	0	0
$467$	0.959348	0.0443933	0.0221967	−	0.999754i	$-0.492934\pi$
$467$	0.959348	0.0443933	0.0221967	+	0.999754i	$0.492934\pi$
$468$	0	0
$469$	−1.19929	−0.0553783
$470$	0	0
$471$	13.8243	0.636989
$472$	0	0
$473$	−39.1411	−1.79971
$474$	0	0
$475$	−2.61323	−0.119903
$476$	0	0
$477$	6.35299	0.290883
$478$	0	0
$479$	29.8621	1.36443	0.682217	−	0.731150i	$-0.261016\pi$
$479$	29.8621	1.36443	0.682217	+	0.731150i	$0.261016\pi$
$480$	0	0
$481$	−6.09962	−0.278119
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	49.2962	2.23842
$486$	0	0
$487$	37.3021	1.69032	0.845160	−	0.534513i	$-0.179505\pi$
$487$	37.3021	1.69032	0.845160	+	0.534513i	$0.179505\pi$
$488$	0	0
$489$	13.7433	0.621493
$490$	0	0
$491$	25.2794	1.14084	0.570422	−	0.821351i	$-0.306780\pi$
$491$	25.2794	1.14084	0.570422	+	0.821351i	$0.306780\pi$
$492$	0	0
$493$	−51.3232	−2.31148
$494$	0	0
$495$	17.2347	0.774642
$496$	0	0
$497$	6.89939	0.309480
$498$	0	0
$499$	−25.0944	−1.12338	−0.561689	−	0.827348i	$-0.689848\pi$
$499$	−25.0944	−1.12338	−0.561689	+	0.827348i	$0.689848\pi$
$500$	0	0
$501$	−0.335294	−0.0149798
$502$	0	0
$503$	−26.2736	−1.17148	−0.585740	−	0.810499i	$-0.699196\pi$
$503$	−26.2736	−1.17148	−0.585740	+	0.810499i	$0.699196\pi$
$504$	0	0
$505$	−56.2202	−2.50176
$506$	0	0
$507$	2.30485	0.102362
$508$	0	0
$509$	−3.86636	−0.171373	−0.0856866	−	0.996322i	$-0.527308\pi$
$509$	−3.86636	−0.171373	−0.0856866	+	0.996322i	$0.527308\pi$
$510$	0	0
$511$	2.73388	0.120940
$512$	0	0
$513$	−0.335294	−0.0148036
$514$	0	0
$515$	−13.2592	−0.584272
$516$	0	0
$517$	−19.7822	−0.870021
$518$	0	0
$519$	0.323531	0.0142014
$520$	0	0
$521$	27.2599	1.19428	0.597138	−	0.802138i	$-0.296304\pi$
$521$	27.2599	1.19428	0.597138	+	0.802138i	$0.296304\pi$
$522$	0	0
$523$	−11.7634	−0.514377	−0.257189	−	0.966361i	$-0.582796\pi$
$523$	−11.7634	−0.514377	−0.257189	+	0.966361i	$0.582796\pi$
$524$	0	0
$525$	−7.79384	−0.340151
$526$	0	0
$527$	8.78301	0.382594
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.08099	−0.0903073
$532$	0	0
$533$	0.165323	0.00716096
$534$	0	0
$535$	9.48899	0.410245
$536$	0	0
$537$	5.06824	0.218711
$538$	0	0
$539$	−4.81840	−0.207543
$540$	0	0
$541$	39.8833	1.71472	0.857359	−	0.514720i	$-0.172104\pi$
$541$	39.8833	1.71472	0.857359	+	0.514720i	$0.172104\pi$
$542$	0	0
$543$	−12.5818	−0.539936
$544$	0	0
$545$	24.5269	1.05062
$546$	0	0
$547$	−12.5504	−0.536615	−0.268307	−	0.963333i	$-0.586464\pi$
$547$	−12.5504	−0.536615	−0.268307	+	0.963333i	$0.586464\pi$
$548$	0	0
$549$	10.9603	0.467773
$550$	0	0
$551$	−2.38442	−0.101580
$552$	0	0
$553$	−16.5582	−0.704125
$554$	0	0
$555$	−5.57685	−0.236724
$556$	0	0
$557$	−37.8951	−1.60567	−0.802833	−	0.596204i	$-0.796675\pi$
$557$	−37.8951	−1.60567	−0.802833	+	0.596204i	$0.796675\pi$
$558$	0	0
$559$	−31.7793	−1.34412
$560$	0	0
$561$	34.7744	1.46817
$562$	0	0
$563$	−18.2498	−0.769139	−0.384570	−	0.923096i	$-0.625650\pi$
$563$	−18.2498	−0.769139	−0.384570	+	0.923096i	$0.625650\pi$
$564$	0	0
$565$	−30.5572	−1.28555
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	15.5076	0.650113	0.325057	−	0.945695i	$-0.394617\pi$
$569$	15.5076	0.650113	0.325057	+	0.945695i	$0.394617\pi$
$570$	0	0
$571$	32.8773	1.37587	0.687936	−	0.725771i	$-0.258517\pi$
$571$	32.8773	1.37587	0.687936	+	0.725771i	$0.258517\pi$
$572$	0	0
$573$	7.06918	0.295319
$574$	0	0
$575$	−7.79384	−0.325026
$576$	0	0
$577$	−24.1384	−1.00489	−0.502446	−	0.864608i	$-0.667567\pi$
$577$	−24.1384	−1.00489	−0.502446	+	0.864608i	$0.667567\pi$
$578$	0	0
$579$	20.4247	0.848822
$580$	0	0
$581$	13.0413	0.541043
$582$	0	0
$583$	−30.6113	−1.26779
$584$	0	0
$585$	13.9931	0.578545
$586$	0	0
$587$	8.53806	0.352404	0.176202	−	0.984354i	$-0.443619\pi$
$587$	8.53806	0.352404	0.176202	+	0.984354i	$0.443619\pi$
$588$	0	0
$589$	0.408049	0.0168134
$590$	0	0
$591$	−6.87836	−0.282938
$592$	0	0
$593$	16.8469	0.691820	0.345910	−	0.938268i	$-0.387570\pi$
$593$	16.8469	0.691820	0.345910	+	0.938268i	$0.387570\pi$
$594$	0	0
$595$	−25.8141	−1.05827
$596$	0	0
$597$	−18.3182	−0.749715
$598$	0	0
$599$	−28.7000	−1.17265	−0.586326	−	0.810075i	$-0.699426\pi$
$599$	−28.7000	−1.17265	−0.586326	+	0.810075i	$0.699426\pi$
$600$	0	0
$601$	10.6840	0.435810	0.217905	−	0.975970i	$-0.430078\pi$
$601$	10.6840	0.435810	0.217905	+	0.975970i	$0.430078\pi$
$602$	0	0
$603$	1.19929	0.0488391
$604$	0	0
$605$	−43.6983	−1.77659
$606$	0	0
$607$	0.751144	0.0304880	0.0152440	−	0.999884i	$-0.495147\pi$
$607$	0.751144	0.0304880	0.0152440	+	0.999884i	$0.495147\pi$
$608$	0	0
$609$	−7.11144	−0.288170
$610$	0	0
$611$	−16.0615	−0.649779
$612$	0	0
$613$	47.1618	1.90485	0.952424	−	0.304777i	$-0.0985820\pi$
$613$	47.1618	1.90485	0.952424	+	0.304777i	$0.0985820\pi$
$614$	0	0
$615$	0.151154	0.00609513
$616$	0	0
$617$	16.8503	0.678366	0.339183	−	0.940720i	$-0.389849\pi$
$617$	16.8503	0.678366	0.339183	+	0.940720i	$0.389849\pi$
$618$	0	0
$619$	−43.0665	−1.73099	−0.865494	−	0.500920i	$-0.832995\pi$
$619$	−43.0665	−1.73099	−0.865494	+	0.500920i	$0.832995\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	9.17139	0.367444
$624$	0	0
$625$	−3.22528	−0.129011
$626$	0	0
$627$	1.61558	0.0645200
$628$	0	0
$629$	−11.2524	−0.448662
$630$	0	0
$631$	41.5881	1.65560	0.827798	−	0.561026i	$-0.189593\pi$
$631$	41.5881	1.65560	0.827798	+	0.561026i	$0.189593\pi$
$632$	0	0
$633$	18.1596	0.721778
$634$	0	0
$635$	56.4134	2.23870
$636$	0	0
$637$	−3.91214	−0.155005
$638$	0	0
$639$	−6.89939	−0.272936
$640$	0	0
$641$	−2.08786	−0.0824655	−0.0412327	−	0.999150i	$-0.513129\pi$
$641$	−2.08786	−0.0824655	−0.0412327	+	0.999150i	$0.513129\pi$
$642$	0	0
$643$	20.3997	0.804486	0.402243	−	0.915533i	$-0.368231\pi$
$643$	20.3997	0.804486	0.402243	+	0.915533i	$0.368231\pi$
$644$	0	0
$645$	−29.0556	−1.14406
$646$	0	0
$647$	21.8750	0.859994	0.429997	−	0.902830i	$-0.358515\pi$
$647$	21.8750	0.859994	0.429997	+	0.902830i	$0.358515\pi$
$648$	0	0
$649$	10.0271	0.393596
$650$	0	0
$651$	1.21699	0.0476976
$652$	0	0
$653$	−18.8788	−0.738784	−0.369392	−	0.929274i	$-0.620434\pi$
$653$	−18.8788	−0.738784	−0.369392	+	0.929274i	$0.620434\pi$
$654$	0	0
$655$	43.2752	1.69090
$656$	0	0
$657$	−2.73388	−0.106659
$658$	0	0
$659$	10.4552	0.407277	0.203638	−	0.979046i	$-0.434723\pi$
$659$	10.4552	0.407277	0.203638	+	0.979046i	$0.434723\pi$
$660$	0	0
$661$	2.21860	0.0862934	0.0431467	−	0.999069i	$-0.486262\pi$
$661$	2.21860	0.0862934	0.0431467	+	0.999069i	$0.486262\pi$
$662$	0	0
$663$	28.2339	1.09651
$664$	0	0
$665$	−1.19929	−0.0465067
$666$	0	0
$667$	−7.11144	−0.275356
$668$	0	0
$669$	25.5421	0.987514
$670$	0	0
$671$	−52.8110	−2.03875
$672$	0	0
$673$	31.8037	1.22594	0.612971	−	0.790105i	$-0.289974\pi$
$673$	31.8037	1.22594	0.612971	+	0.790105i	$0.289974\pi$
$674$	0	0
$675$	7.79384	0.299985
$676$	0	0
$677$	39.2001	1.50658	0.753291	−	0.657687i	$-0.228465\pi$
$677$	39.2001	1.50658	0.753291	+	0.657687i	$0.228465\pi$
$678$	0	0
$679$	13.7820	0.528906
$680$	0	0
$681$	16.7423	0.641567
$682$	0	0
$683$	−22.1242	−0.846558	−0.423279	−	0.905999i	$-0.639121\pi$
$683$	−22.1242	−0.846558	−0.423279	+	0.905999i	$0.639121\pi$
$684$	0	0
$685$	77.6693	2.96759
$686$	0	0
$687$	26.5504	1.01296
$688$	0	0
$689$	−24.8538	−0.946854
$690$	0	0
$691$	−22.1115	−0.841161	−0.420580	−	0.907255i	$-0.638174\pi$
$691$	−22.1115	−0.841161	−0.420580	+	0.907255i	$0.638174\pi$
$692$	0	0
$693$	4.81840	0.183036
$694$	0	0
$695$	−26.1283	−0.991104
$696$	0	0
$697$	0.304983	0.0115521
$698$	0	0
$699$	5.76692	0.218125
$700$	0	0
$701$	−33.0353	−1.24773	−0.623864	−	0.781533i	$-0.714438\pi$
$701$	−33.0353	−1.24773	−0.623864	+	0.781533i	$0.714438\pi$
$702$	0	0
$703$	−0.522774	−0.0197168
$704$	0	0
$705$	−14.6849	−0.553067
$706$	0	0
$707$	−15.7178	−0.591128
$708$	0	0
$709$	−11.9265	−0.447909	−0.223954	−	0.974600i	$-0.571897\pi$
$709$	−11.9265	−0.447909	−0.223954	+	0.974600i	$0.571897\pi$
$710$	0	0
$711$	16.5582	0.620980
$712$	0	0
$713$	1.21699	0.0455767
$714$	0	0
$715$	−67.4245	−2.52153
$716$	0	0
$717$	23.8023	0.888914
$718$	0	0
$719$	−39.0840	−1.45759	−0.728793	−	0.684734i	$-0.759918\pi$
$719$	−39.0840	−1.45759	−0.728793	+	0.684734i	$0.759918\pi$
$720$	0	0
$721$	−3.70697	−0.138055
$722$	0	0
$723$	−12.1031	−0.450121
$724$	0	0
$725$	55.4254	2.05845
$726$	0	0
$727$	27.2170	1.00942	0.504711	−	0.863288i	$-0.331599\pi$
$727$	27.2170	1.00942	0.504711	+	0.863288i	$0.331599\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−58.6254	−2.16834
$732$	0	0
$733$	47.6585	1.76031	0.880153	−	0.474691i	$-0.157440\pi$
$733$	47.6585	1.76031	0.880153	+	0.474691i	$0.157440\pi$
$734$	0	0
$735$	−3.57685	−0.131934
$736$	0	0
$737$	−5.77868	−0.212861
$738$	0	0
$739$	16.1341	0.593502	0.296751	−	0.954955i	$-0.404097\pi$
$739$	16.1341	0.593502	0.296751	+	0.954955i	$0.404097\pi$
$740$	0	0
$741$	1.31172	0.0481871
$742$	0	0
$743$	−25.2347	−0.925771	−0.462885	−	0.886418i	$-0.653186\pi$
$743$	−25.2347	−0.925771	−0.462885	+	0.886418i	$0.653186\pi$
$744$	0	0
$745$	−12.1266	−0.444284
$746$	0	0
$747$	−13.0413	−0.477155
$748$	0	0
$749$	2.65289	0.0969346
$750$	0	0
$751$	2.57586	0.0939945	0.0469973	−	0.998895i	$-0.485035\pi$
$751$	2.57586	0.0939945	0.0469973	+	0.998895i	$0.485035\pi$
$752$	0	0
$753$	−7.80993	−0.284610
$754$	0	0
$755$	67.4126	2.45339
$756$	0	0
$757$	−36.1086	−1.31239	−0.656194	−	0.754592i	$-0.727835\pi$
$757$	−36.1086	−1.31239	−0.656194	+	0.754592i	$0.727835\pi$
$758$	0	0
$759$	4.81840	0.174897
$760$	0	0
$761$	0.687163	0.0249097	0.0124548	−	0.999922i	$-0.496035\pi$
$761$	0.687163	0.0249097	0.0124548	+	0.999922i	$0.496035\pi$
$762$	0	0
$763$	6.85713	0.248245
$764$	0	0
$765$	25.8141	0.933310
$766$	0	0
$767$	8.14113	0.293959
$768$	0	0
$769$	31.1426	1.12303	0.561515	−	0.827467i	$-0.310219\pi$
$769$	31.1426	1.12303	0.561515	+	0.827467i	$0.310219\pi$
$770$	0	0
$771$	−2.26024	−0.0814004
$772$	0	0
$773$	44.8425	1.61287	0.806437	−	0.591320i	$-0.201393\pi$
$773$	44.8425	1.61287	0.806437	+	0.591320i	$0.201393\pi$
$774$	0	0
$775$	−9.48503	−0.340712
$776$	0	0
$777$	−1.55915	−0.0559343
$778$	0	0
$779$	0.0141692	0.000507665 0
$780$	0	0
$781$	33.2440	1.18957
$782$	0	0
$783$	7.11144	0.254142
$784$	0	0
$785$	−49.4474	−1.76485
$786$	0	0
$787$	12.3412	0.439917	0.219959	−	0.975509i	$-0.429408\pi$
$787$	12.3412	0.439917	0.219959	+	0.975509i	$0.429408\pi$
$788$	0	0
$789$	−20.0168	−0.712616
$790$	0	0
$791$	−8.54306	−0.303756
$792$	0	0
$793$	−42.8782	−1.52265
$794$	0	0
$795$	−22.7237	−0.805926
$796$	0	0
$797$	52.4818	1.85900	0.929500	−	0.368823i	$-0.120239\pi$
$797$	52.4818	1.85900	0.929500	+	0.368823i	$0.120239\pi$
$798$	0	0
$799$	−29.6297	−1.04823
$800$	0	0
$801$	−9.17139	−0.324055
$802$	0	0
$803$	13.1729	0.464863
$804$	0	0
$805$	−3.57685	−0.126067
$806$	0	0
$807$	30.1730	1.06214
$808$	0	0
$809$	−33.6408	−1.18275	−0.591373	−	0.806398i	$-0.701414\pi$
$809$	−33.6408	−1.18275	−0.591373	+	0.806398i	$0.701414\pi$
$810$	0	0
$811$	−10.6500	−0.373972	−0.186986	−	0.982363i	$-0.559872\pi$
$811$	−10.6500	−0.373972	−0.186986	+	0.982363i	$0.559872\pi$
$812$	0	0
$813$	19.5735	0.686473
$814$	0	0
$815$	−49.1577	−1.72192
$816$	0	0
$817$	−2.72368	−0.0952893
$818$	0	0
$819$	3.91214	0.136701
$820$	0	0
$821$	50.3310	1.75656	0.878282	−	0.478142i	$-0.158690\pi$
$821$	50.3310	1.75656	0.878282	+	0.478142i	$0.158690\pi$
$822$	0	0
$823$	−18.1916	−0.634120	−0.317060	−	0.948405i	$-0.602696\pi$
$823$	−18.1916	−0.634120	−0.317060	+	0.948405i	$0.602696\pi$
$824$	0	0
$825$	−37.5538	−1.30746
$826$	0	0
$827$	33.1298	1.15203	0.576017	−	0.817438i	$-0.304606\pi$
$827$	33.1298	1.15203	0.576017	+	0.817438i	$0.304606\pi$
$828$	0	0
$829$	−9.34555	−0.324584	−0.162292	−	0.986743i	$-0.551889\pi$
$829$	−9.34555	−0.324584	−0.162292	+	0.986743i	$0.551889\pi$
$830$	0	0
$831$	−22.1140	−0.767125
$832$	0	0
$833$	−7.21699	−0.250054
$834$	0	0
$835$	1.19929	0.0415033
$836$	0	0
$837$	−1.21699	−0.0420653
$838$	0	0
$839$	11.4482	0.395236	0.197618	−	0.980279i	$-0.436679\pi$
$839$	11.4482	0.395236	0.197618	+	0.980279i	$0.436679\pi$
$840$	0	0
$841$	21.5725	0.743880
$842$	0	0
$843$	−21.7424	−0.748847
$844$	0	0
$845$	−8.24409	−0.283605
$846$	0	0
$847$	−12.2170	−0.419781
$848$	0	0
$849$	5.59788	0.192119
$850$	0	0
$851$	−1.55915	−0.0534470
$852$	0	0
$853$	−53.2336	−1.82269	−0.911343	−	0.411648i	$-0.864953\pi$
$853$	−53.2336	−1.82269	−0.911343	+	0.411648i	$0.864953\pi$
$854$	0	0
$855$	1.19929	0.0410150
$856$	0	0
$857$	−7.95025	−0.271575	−0.135788	−	0.990738i	$-0.543357\pi$
$857$	−7.95025	−0.271575	−0.135788	+	0.990738i	$0.543357\pi$
$858$	0	0
$859$	−16.1664	−0.551592	−0.275796	−	0.961216i	$-0.588941\pi$
$859$	−16.1664	−0.551592	−0.275796	+	0.961216i	$0.588941\pi$
$860$	0	0
$861$	0.0422591	0.00144019
$862$	0	0
$863$	−25.2500	−0.859519	−0.429760	−	0.902943i	$-0.641402\pi$
$863$	−25.2500	−0.859519	−0.429760	+	0.902943i	$0.641402\pi$
$864$	0	0
$865$	−1.15722	−0.0393467
$866$	0	0
$867$	35.0850	1.19155
$868$	0	0
$869$	−79.7839	−2.70648
$870$	0	0
$871$	−4.69181	−0.158976
$872$	0	0
$873$	−13.7820	−0.466451
$874$	0	0
$875$	9.99313	0.337830
$876$	0	0
$877$	−2.08452	−0.0703892	−0.0351946	−	0.999380i	$-0.511205\pi$
$877$	−2.08452	−0.0703892	−0.0351946	+	0.999380i	$0.511205\pi$
$878$	0	0
$879$	−14.6706	−0.494827
$880$	0	0
$881$	7.69769	0.259342	0.129671	−	0.991557i	$-0.458608\pi$
$881$	7.69769	0.259342	0.129671	+	0.991557i	$0.458608\pi$
$882$	0	0
$883$	57.9600	1.95051	0.975255	−	0.221083i	$-0.0709592\pi$
$883$	57.9600	1.95051	0.975255	+	0.221083i	$0.0709592\pi$
$884$	0	0
$885$	7.44339	0.250207
$886$	0	0
$887$	39.4481	1.32454	0.662269	−	0.749266i	$-0.269594\pi$
$887$	39.4481	1.32454	0.662269	+	0.749266i	$0.269594\pi$
$888$	0	0
$889$	15.7718	0.528970
$890$	0	0
$891$	−4.81840	−0.161423
$892$	0	0
$893$	−1.37657	−0.0460651
$894$	0	0
$895$	−18.1283	−0.605963
$896$	0	0
$897$	3.91214	0.130623
$898$	0	0
$899$	−8.65455	−0.288645
$900$	0	0
$901$	−45.8495	−1.52747
$902$	0	0
$903$	−8.12325	−0.270325
$904$	0	0
$905$	45.0032	1.49596
$906$	0	0
$907$	−30.4448	−1.01090	−0.505452	−	0.862855i	$-0.668674\pi$
$907$	−30.4448	−1.01090	−0.505452	+	0.862855i	$0.668674\pi$
$908$	0	0
$909$	15.7178	0.521326
$910$	0	0
$911$	45.3234	1.50163	0.750815	−	0.660513i	$-0.229661\pi$
$911$	45.3234	1.50163	0.750815	+	0.660513i	$0.229661\pi$
$912$	0	0
$913$	62.8381	2.07964
$914$	0	0
$915$	−39.2033	−1.29602
$916$	0	0
$917$	12.0987	0.399534
$918$	0	0
$919$	47.2201	1.55765	0.778824	−	0.627243i	$-0.215817\pi$
$919$	47.2201	1.55765	0.778824	+	0.627243i	$0.215817\pi$
$920$	0	0
$921$	9.07604	0.299066
$922$	0	0
$923$	26.9914	0.888433
$924$	0	0
$925$	12.1518	0.399548
$926$	0	0
$927$	3.70697	0.121753
$928$	0	0
$929$	−42.4145	−1.39157	−0.695787	−	0.718248i	$-0.744944\pi$
$929$	−42.4145	−1.39157	−0.695787	+	0.718248i	$0.744944\pi$
$930$	0	0
$931$	−0.335294	−0.0109888
$932$	0	0
$933$	−4.10221	−0.134300
$934$	0	0
$935$	−124.383	−4.06774
$936$	0	0
$937$	−2.68451	−0.0876991	−0.0438495	−	0.999038i	$-0.513962\pi$
$937$	−2.68451	−0.0876991	−0.0438495	+	0.999038i	$0.513962\pi$
$938$	0	0
$939$	2.09628	0.0684095
$940$	0	0
$941$	−42.9923	−1.40151	−0.700755	−	0.713402i	$-0.747153\pi$
$941$	−42.9923	−1.40151	−0.700755	+	0.713402i	$0.747153\pi$
$942$	0	0
$943$	0.0422591	0.00137615
$944$	0	0
$945$	3.57685	0.116355
$946$	0	0
$947$	−27.2997	−0.887122	−0.443561	−	0.896244i	$-0.646285\pi$
$947$	−27.2997	−0.887122	−0.443561	+	0.896244i	$0.646285\pi$
$948$	0	0
$949$	10.6953	0.347185
$950$	0	0
$951$	10.9063	0.353660
$952$	0	0
$953$	−36.8015	−1.19212	−0.596059	−	0.802941i	$-0.703268\pi$
$953$	−36.8015	−1.19212	−0.596059	+	0.802941i	$0.703268\pi$
$954$	0	0
$955$	−25.2854	−0.818215
$956$	0	0
$957$	−34.2658	−1.10765
$958$	0	0
$959$	21.7145	0.701196
$960$	0	0
$961$	−29.5189	−0.952224
$962$	0	0
$963$	−2.65289	−0.0854882
$964$	0	0
$965$	−73.0561	−2.35176
$966$	0	0
$967$	−38.9270	−1.25181	−0.625904	−	0.779900i	$-0.715270\pi$
$967$	−38.9270	−1.25181	−0.625904	+	0.779900i	$0.715270\pi$
$968$	0	0
$969$	2.41981	0.0777356
$970$	0	0
$971$	−56.2523	−1.80522	−0.902612	−	0.430456i	$-0.858353\pi$
$971$	−56.2523	−1.80522	−0.902612	+	0.430456i	$0.858353\pi$
$972$	0	0
$973$	−7.30485	−0.234183
$974$	0	0
$975$	−30.4906	−0.976481
$976$	0	0
$977$	43.0147	1.37616	0.688082	−	0.725633i	$-0.258453\pi$
$977$	43.0147	1.37616	0.688082	+	0.725633i	$0.258453\pi$
$978$	0	0
$979$	44.1914	1.41236
$980$	0	0
$981$	−6.85713	−0.218931
$982$	0	0
$983$	−32.3824	−1.03284	−0.516420	−	0.856336i	$-0.672736\pi$
$983$	−32.3824	−1.03284	−0.516420	+	0.856336i	$0.672736\pi$
$984$	0	0
$985$	24.6028	0.783911
$986$	0	0
$987$	−4.10555	−0.130681
$988$	0	0
$989$	−8.12325	−0.258304
$990$	0	0
$991$	−16.9957	−0.539885	−0.269943	−	0.962876i	$-0.587005\pi$
$991$	−16.9957	−0.539885	−0.269943	+	0.962876i	$0.587005\pi$
$992$	0	0
$993$	29.6368	0.940495
$994$	0	0
$995$	65.5215	2.07717
$996$	0	0
$997$	17.6882	0.560192	0.280096	−	0.959972i	$-0.409634\pi$
$997$	17.6882	0.560192	0.280096	+	0.959972i	$0.409634\pi$
$998$	0	0
$999$	1.55915	0.0493294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.2.a.t.1.1	✓	4
4.3	odd	2		7728.2.a.bz.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.t.1.1	✓	4	1.1	even	1	trivial
7728.2.a.bz.1.1		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.57685 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.57685 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.81840 q^{11} -3.91214 q^{13} -3.57685 q^{15} -7.21699 q^{17} -0.335294 q^{19} -1.00000 q^{21} -1.00000 q^{23} +7.79384 q^{25} +1.00000 q^{27} +7.11144 q^{29} -1.21699 q^{31} -4.81840 q^{33} +3.57685 q^{35} +1.55915 q^{37} -3.91214 q^{39} -0.0422591 q^{41} +8.12325 q^{43} -3.57685 q^{45} +4.10555 q^{47} +1.00000 q^{49} -7.21699 q^{51} +6.35299 q^{53} +17.2347 q^{55} -0.335294 q^{57} -2.08099 q^{59} +10.9603 q^{61} -1.00000 q^{63} +13.9931 q^{65} +1.19929 q^{67} -1.00000 q^{69} -6.89939 q^{71} -2.73388 q^{73} +7.79384 q^{75} +4.81840 q^{77} +16.5582 q^{79} +1.00000 q^{81} -13.0413 q^{83} +25.8141 q^{85} +7.11144 q^{87} -9.17139 q^{89} +3.91214 q^{91} -1.21699 q^{93} +1.19929 q^{95} -13.7820 q^{97} -4.81840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{21} - 4 q^{23} + 4 q^{27} + 11 q^{29} + 14 q^{31} + 2 q^{35} + 13 q^{37} + 2 q^{39} + 7 q^{41} + 12 q^{43} - 2 q^{45} + 15 q^{47} + 4 q^{49} - 10 q^{51} + q^{53} + 31 q^{55} + 4 q^{57} + 5 q^{59} + 3 q^{61} - 4 q^{63} + 25 q^{65} + 5 q^{67} - 4 q^{69} + 5 q^{71} - 6 q^{73} + 26 q^{79} + 4 q^{81} + 2 q^{83} + 5 q^{85} + 11 q^{87} + 7 q^{89} - 2 q^{91} + 14 q^{93} + 5 q^{95} - 27 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^13 - 2 * q^15 - 10 * q^17 + 4 * q^19 - 4 * q^21 - 4 * q^23 + 4 * q^27 + 11 * q^29 + 14 * q^31 + 2 * q^35 + 13 * q^37 + 2 * q^39 + 7 * q^41 + 12 * q^43 - 2 * q^45 + 15 * q^47 + 4 * q^49 - 10 * q^51 + q^53 + 31 * q^55 + 4 * q^57 + 5 * q^59 + 3 * q^61 - 4 * q^63 + 25 * q^65 + 5 * q^67 - 4 * q^69 + 5 * q^71 - 6 * q^73 + 26 * q^79 + 4 * q^81 + 2 * q^83 + 5 * q^85 + 11 * q^87 + 7 * q^89 - 2 * q^91 + 14 * q^93 + 5 * q^95 - 27 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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