LMFDB - Embedded newform 39.4.f.a.8.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,4,Mod(5,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("39.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	39.f (of order $4$, degree $2$, minimal)

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:	no
Analytic conductor:	$2.30107449022$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			8.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	39.8
Dual form			39.4.f.a.5.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} +8.00000i q^{4} +(5.58846 - 5.58846i) q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} +8.00000i q^{4} +(5.58846 - 5.58846i) q^{7} +27.0000 q^{9} +41.5692i q^{12} +(-31.1769 - 35.0000i) q^{13} -64.0000 q^{16} +(-105.942 - 105.942i) q^{19} +(29.0385 - 29.0385i) q^{21} +125.000i q^{25} +140.296 q^{27} +(44.7077 + 44.7077i) q^{28} +(76.0577 + 76.0577i) q^{31} +216.000i q^{36} +(273.238 - 273.238i) q^{37} +(-162.000 - 181.865i) q^{39} +218.238i q^{43} -332.554 q^{48} +280.538i q^{49} +(280.000 - 249.415i) q^{52} +(-550.492 - 550.492i) q^{57} -935.307 q^{61} +(150.888 - 150.888i) q^{63} -512.000i q^{64} +(767.358 + 767.358i) q^{67} +(-782.061 + 782.061i) q^{73} +649.519i q^{75} +(847.538 - 847.538i) q^{76} +1091.19 q^{79} +729.000 q^{81} +(232.308 + 232.308i) q^{84} +(-369.827 - 21.3651i) q^{91} +(395.207 + 395.207i) q^{93} +(-1350.89 - 1350.89i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 40 q^{7} + 108 q^{9}+O(q^{10}) $ 4 * q - 40 * q^7 + 108 * q^9	$ 4 q - 40 q^{7} + 108 q^{9} - 256 q^{16} - 112 q^{19} + 324 q^{21} - 320 q^{28} + 616 q^{31} + 220 q^{37} - 648 q^{39} + 1120 q^{52} - 1620 q^{57} - 1080 q^{63} + 1760 q^{67} - 2380 q^{73} + 896 q^{76} + 2916 q^{81} + 2592 q^{84} - 544 q^{91} - 1620 q^{93} - 2660 q^{97}+O(q^{100}) $ 4 * q - 40 * q^7 + 108 * q^9 - 256 * q^16 - 112 * q^19 + 324 * q^21 - 320 * q^28 + 616 * q^31 + 220 * q^37 - 648 * q^39 + 1120 * q^52 - 1620 * q^57 - 1080 * q^63 + 1760 * q^67 - 2380 * q^73 + 896 * q^76 + 2916 * q^81 + 2592 * q^84 - 544 * q^91 - 1620 * q^93 - 2660 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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		−0.707107	−	0.707107i	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$	5.19615			1.00000
$3$	5.19615			1.00000
$4$			8.00000i			1.00000i
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$	5.58846	−	5.58846i	0.301748	−	0.301748i	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	5.58846	−	5.58846i	0.301748	−	0.301748i	0.841698	+	0.539949i	$0.181557\pi$
$8$		0			0
$9$	27.0000			1.00000
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$			41.5692i			1.00000i
$13$	−31.1769	−	35.0000i	−0.665148	−	0.746712i
$13$	−31.1769	−	35.0000i	−0.665148	−	0.746712i
$14$		0			0
$15$		0			0
$16$	−64.0000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−105.942	−	105.942i	−1.27920	−	1.27920i	−0.941115	−	0.338086i	$-0.890220\pi$
$19$	−105.942	−	105.942i	−1.27920	−	1.27920i	−0.338086	−	0.941115i	$-0.609780\pi$
$20$		0			0
$21$	29.0385	−	29.0385i	0.301748	−	0.301748i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$			125.000i			1.00000i
$26$		0			0
$27$	140.296			1.00000
$28$	44.7077	+	44.7077i	0.301748	+	0.301748i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	76.0577	+	76.0577i	0.440657	+	0.440657i	0.892233	−	0.451576i	$-0.149138\pi$
$31$	76.0577	+	76.0577i	0.440657	+	0.440657i	−0.451576	+	0.892233i	$0.649138\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$			216.000i			1.00000i
$37$	273.238	−	273.238i	1.21406	−	1.21406i	0.244377	−	0.969680i	$-0.421417\pi$
$37$	273.238	−	273.238i	1.21406	−	1.21406i	0.969680	−	0.244377i	$-0.0785834\pi$
$38$		0			0
$39$	−162.000	−	181.865i	−0.665148	−	0.746712i
$40$		0			0
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$			218.238i			0.773978i	0.922084	+	0.386989i	$0.126485\pi$
$43$			218.238i			0.773978i	−0.922084	+	0.386989i	$0.873515\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	−332.554			−1.00000
$49$			280.538i			0.817896i
$50$		0			0
$51$		0			0
$52$	280.000	−	249.415i	0.746712	−	0.665148i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−550.492	−	550.492i	−1.27920	−	1.27920i
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−935.307			−1.96318			−0.981589	−	0.191006i	$-0.938825\pi$
$61$	−935.307			−1.96318			−0.981589	+	0.191006i	$0.938825\pi$
$62$		0			0
$63$	150.888	−	150.888i	0.301748	−	0.301748i
$64$		−	512.000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	767.358	+	767.358i	1.39922	+	1.39922i	0.802307	+	0.596912i	$0.203606\pi$
$67$	767.358	+	767.358i	1.39922	+	1.39922i	0.596912	+	0.802307i	$0.296394\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$		0			0
$73$	−782.061	+	782.061i	−1.25388	+	1.25388i	−0.299916	+	0.953966i	$0.596959\pi$
$73$	−782.061	+	782.061i	−1.25388	+	1.25388i	−0.953966	+	0.299916i	$0.903041\pi$
$74$		0			0
$75$			649.519i			1.00000i
$76$	847.538	−	847.538i	1.27920	−	1.27920i
$77$		0			0
$78$		0			0
$79$	1091.19			1.55403			0.777017	−	0.629480i	$-0.216732\pi$
$79$	1091.19			1.55403			0.777017	+	0.629480i	$0.216732\pi$
$80$		0			0
$81$	729.000			1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$	232.308	+	232.308i	0.301748	+	0.301748i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$	−369.827	−	21.3651i	−0.426026	−	0.0246118i
$92$		0			0
$93$	395.207	+	395.207i	0.440657	+	0.440657i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1350.89	−	1350.89i	−1.41404	−	1.41404i	−0.717957	−	0.696088i	$-0.754922\pi$
$97$	−1350.89	−	1350.89i	−1.41404	−	1.41404i	−0.696088	−	0.717957i	$-0.745078\pi$
$98$		0			0
$99$		0			0
$100$	−1000.00			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			1028.84i			0.984218i	0.870534	+	0.492109i	$0.163774\pi$
$103$			1028.84i			0.984218i	−0.870534	+	0.492109i	$0.836226\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			1122.37i			1.00000i
$109$	−768.192	−	768.192i	−0.675041	−	0.675041i	0.283833	−	0.958874i	$-0.408394\pi$
$109$	−768.192	−	768.192i	−0.675041	−	0.675041i	−0.958874	+	0.283833i	$0.908394\pi$
$110$		0			0
$111$	1419.79	−	1419.79i	1.21406	−	1.21406i
$112$	−357.661	+	357.661i	−0.301748	+	0.301748i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−841.777	−	945.000i	−0.665148	−	0.746712i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	1331.00i		−	1.00000i
$122$		0			0
$123$		0			0
$124$	−608.462	+	608.462i	−0.440657	+	0.440657i
$125$		0			0
$126$		0			0
$127$			380.000i			0.265508i	0.991149	+	0.132754i	$0.0423821\pi$
$127$			380.000i			0.265508i	−0.991149	+	0.132754i	$0.957618\pi$
$128$		0			0
$129$			1134.00i			0.773978i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−1184.11			−0.771994
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	2576.00			1.57190			0.785948	−	0.618293i	$-0.212175\pi$
$139$	2576.00			1.57190			0.785948	+	0.618293i	$0.212175\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1728.00			−1.00000
$145$		0			0
$146$		0			0
$147$			1457.72i			0.817896i
$148$	2185.91	+	2185.91i	1.21406	+	1.21406i
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$	2510.79	−	2510.79i	1.35315	−	1.35315i	0.471027	−	0.882119i	$-0.343883\pi$
$151$	2510.79	−	2510.79i	1.35315	−	1.35315i	0.882119	−	0.471027i	$-0.156117\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1454.92	−	1296.00i	0.746712	−	0.665148i
$157$	3850.00			1.95709			0.978546	−	0.206028i	$-0.0660539\pi$
$157$	3850.00			1.95709			0.978546	+	0.206028i	$0.0660539\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2900.31	+	2900.31i	−1.39368	+	1.39368i	−0.576783	+	0.816897i	$0.695692\pi$
$163$	−2900.31	+	2900.31i	−1.39368	+	1.39368i	−0.816897	+	0.576783i	$0.804308\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	−253.000	+	2182.38i	−0.115157	+	0.993347i
$170$		0			0
$171$	−2860.44	−	2860.44i	−1.27920	−	1.27920i
$172$	−1745.91			−0.773978
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	698.557	+	698.557i	0.301748	+	0.301748i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		−	3429.46i		−	1.40834i	−0.710031	−	0.704171i	$-0.751319\pi$
$181$		−	3429.46i		−	1.40834i	0.710031	−	0.704171i	$-0.248681\pi$
$182$		0			0
$183$	−4860.00			−1.96318
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	784.039	−	784.039i	0.301748	−	0.301748i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	2660.43i		−	1.00000i
$193$	−2043.86	+	2043.86i	−0.762281	+	0.762281i	−0.976734	−	0.214453i	$-0.931203\pi$
$193$	−2043.86	+	2043.86i	−0.762281	+	0.762281i	0.214453	+	0.976734i	$0.431203\pi$
$194$		0			0
$195$		0			0
$196$	−2244.31			−0.817896
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		−	5236.00i		−	1.86518i	−0.360942	−	0.932588i	$-0.617545\pi$
$199$		−	5236.00i		−	1.86518i	0.360942	−	0.932588i	$-0.382455\pi$
$200$		0			0
$201$	3987.31	+	3987.31i	1.39922	+	1.39922i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1995.32	+	2240.00i	0.665148	+	0.746712i
$209$		0			0
$210$		0			0
$211$	1091.19			0.356023			0.178011	−	0.984028i	$-0.443034\pi$
$211$	1091.19			0.356023			0.178011	+	0.984028i	$0.443034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	850.091			0.265935
$218$		0			0
$219$	−4063.71	+	4063.71i	−1.25388	+	1.25388i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1305.04	−	1305.04i	−0.391893	−	0.391893i	0.483469	−	0.875362i	$-0.339377\pi$
$223$	−1305.04	−	1305.04i	−0.391893	−	0.391893i	−0.875362	+	0.483469i	$0.839377\pi$
$224$		0			0
$225$			3375.00i			1.00000i
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$	4403.94	−	4403.94i	1.27920	−	1.27920i
$229$	417.038	−	417.038i	0.120343	−	0.120343i	−0.644370	−	0.764714i	$-0.722880\pi$
$229$	417.038	−	417.038i	0.120343	−	0.120343i	0.764714	+	0.644370i	$0.222880\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	5670.00			1.55403
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−3065.46	+	3065.46i	−0.819352	+	0.819352i	−0.986014	−	0.166662i	$-0.946701\pi$
$241$	−3065.46	+	3065.46i	−0.819352	+	0.819352i	0.166662	+	0.986014i	$0.446701\pi$
$242$		0			0
$243$	3788.00			1.00000
$244$		−	7482.46i		−	1.96318i
$245$		0			0
$246$		0			0
$247$	−405.026	+	7010.93i	−0.104337	+	1.80605i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	1207.11	+	1207.11i	0.301748	+	0.301748i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4096.00			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		−	3053.96i		−	0.732679i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−6138.86	+	6138.86i	−1.39922	+	1.39922i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−4036.71	+	4036.71i	−0.904844	+	0.904844i	−0.995850	−	0.0910064i	$-0.970992\pi$
$271$	−4036.71	+	4036.71i	−0.904844	+	0.904844i	0.0910064	+	0.995850i	$0.470992\pi$
$272$		0			0
$273$	−1921.68	−	111.017i	−0.426026	−	0.0246118i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	8293.06i		−	1.79885i	−0.437074	−	0.899425i	$-0.643985\pi$
$277$		−	8293.06i		−	1.79885i	0.437074	−	0.899425i	$-0.356015\pi$
$278$		0			0
$279$	2053.56	+	2053.56i	0.440657	+	0.440657i
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$		−	5600.00i		−	1.17627i	−0.808761	−	0.588137i	$-0.799862\pi$
$283$		−	5600.00i		−	1.17627i	0.808761	−	0.588137i	$-0.200138\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4913.00			−1.00000
$290$		0			0
$291$	−7019.44	−	7019.44i	−1.41404	−	1.41404i
$292$	−6256.49	−	6256.49i	−1.25388	−	1.25388i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−5196.15			−1.00000
$301$	1219.62	+	1219.62i	0.233547	+	0.233547i
$302$		0			0
$303$		0			0
$304$	6780.31	+	6780.31i	1.27920	+	1.27920i
$305$		0			0
$306$		0			0
$307$	4524.99	−	4524.99i	0.841221	−	0.841221i	−0.147797	−	0.989018i	$-0.547218\pi$
$307$	4524.99	−	4524.99i	0.841221	−	0.841221i	0.989018	+	0.147797i	$0.0472182\pi$
$308$		0			0
$309$			5346.00i			0.984218i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	4738.89			0.855776			0.427888	−	0.903832i	$-0.359258\pi$
$313$	4738.89			0.855776			0.427888	+	0.903832i	$0.359258\pi$
$314$		0			0
$315$		0			0
$316$			8729.54i			1.55403i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$			5832.00i			1.00000i
$325$	4375.00	−	3897.11i	0.746712	−	0.665148i
$326$		0			0
$327$	−3991.64	−	3991.64i	−0.675041	−	0.675041i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5505.56	+	5505.56i	0.914238	+	0.914238i	0.996602	−	0.0823644i	$-0.0262471\pi$
$331$	5505.56	+	5505.56i	0.914238	+	0.914238i	−0.0823644	+	0.996602i	$0.526247\pi$
$332$		0			0
$333$	7377.44	−	7377.44i	1.21406	−	1.21406i
$334$		0			0
$335$		0			0
$336$	−1858.46	+	1858.46i	−0.301748	+	0.301748i
$337$			4930.00i			0.796897i	0.917191	+	0.398448i	$0.130451\pi$
$337$			4930.00i			0.796897i	−0.917191	+	0.398448i	$0.869549\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	3484.62	+	3484.62i	0.548547	+	0.548547i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	8607.04	−	8607.04i	1.32013	−	1.32013i	0.406456	−	0.913670i	$-0.366764\pi$
$349$	8607.04	−	8607.04i	1.32013	−	1.32013i	0.913670	−	0.406456i	$-0.133236\pi$
$350$		0			0
$351$	−4374.00	−	4910.36i	−0.665148	−	0.746712i
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$			15588.5i			2.27271i
$362$		0			0
$363$		−	6916.08i		−	1.00000i
$364$	170.921	−	2958.61i	0.0246118	−	0.426026i
$365$		0			0
$366$		0			0
$367$	−4340.00			−0.617292			−0.308646	−	0.951177i	$-0.599876\pi$
$367$	−4340.00			−0.617292			−0.308646	+	0.951177i	$0.599876\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3161.66	+	3161.66i	−0.440657	+	0.440657i
$373$	−7420.11			−1.03002			−0.515011	−	0.857183i	$-0.672212\pi$
$373$	−7420.11			−1.03002			−0.515011	+	0.857183i	$0.672212\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1709.56	+	1709.56i	0.231699	+	0.231699i	0.813402	−	0.581702i	$-0.197613\pi$
$379$	1709.56	+	1709.56i	0.231699	+	0.231699i	−0.581702	+	0.813402i	$0.697613\pi$
$380$		0			0
$381$			1974.54i			0.265508i
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			5892.44i			0.773978i
$388$	10807.1	−	10807.1i	1.41404	−	1.41404i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−8482.76	+	8482.76i	−1.07239	+	1.07239i	−0.0752196	+	0.997167i	$0.523966\pi$
$397$	−8482.76	+	8482.76i	−1.07239	+	1.07239i	−0.997167	+	0.0752196i	$0.976034\pi$
$398$		0			0
$399$	−6152.81			−0.771994
$400$		−	8000.00i		−	1.00000i
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$	290.775	−	5033.26i	0.0359418	−	0.622146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−11293.7	−	11293.7i	−1.36537	−	1.36537i	−0.866914	−	0.498458i	$-0.833900\pi$
$409$	−11293.7	−	11293.7i	−1.36537	−	1.36537i	−0.498458	−	0.866914i	$-0.666100\pi$
$410$		0			0
$411$		0			0
$412$	−8230.71			−0.984218
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	13385.3			1.57190
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−9660.19	−	9660.19i	−1.11831	−	1.11831i	−0.991989	−	0.126322i	$-0.959683\pi$
$421$	−9660.19	−	9660.19i	−1.11831	−	1.11831i	−0.126322	−	0.991989i	$-0.540317\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−5226.93	+	5226.93i	−0.592386	+	0.592386i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	−8978.95			−1.00000
$433$			2590.00i			0.287454i	0.989617	+	0.143727i	$0.0459087\pi$
$433$			2590.00i			0.287454i	−0.989617	+	0.143727i	$0.954091\pi$
$434$		0			0
$435$		0			0
$436$	6145.54	−	6145.54i	0.675041	−	0.675041i
$437$		0			0
$438$		0			0
$439$			10756.0i			1.16938i	0.811257	+	0.584690i	$0.198784\pi$
$439$			10756.0i			1.16938i	−0.811257	+	0.584690i	$0.801216\pi$
$440$		0			0
$441$			7574.53i			0.817896i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	11358.3	+	11358.3i	1.21406	+	1.21406i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−2861.29	−	2861.29i	−0.301748	−	0.301748i
$449$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$449$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	13046.4	−	13046.4i	1.35315	−	1.35315i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	13775.1	+	13775.1i	1.41000	+	1.41000i	0.759514	+	0.650491i	$0.225437\pi$
$457$	13775.1	+	13775.1i	1.41000	+	1.41000i	0.650491	+	0.759514i	$0.274563\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	−11090.3	+	11090.3i	−1.11320	+	1.11320i	−0.120482	+	0.992716i	$0.538444\pi$
$463$	−11090.3	+	11090.3i	−1.11320	+	1.11320i	−0.992716	+	0.120482i	$0.961556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	7560.00	−	6734.21i	0.746712	−	0.665148i
$469$	8576.69			0.844424
$470$		0			0
$471$	20005.2			1.95709
$472$		0			0
$473$		0			0
$474$		0			0
$475$	13242.8	−	13242.8i	1.27920	−	1.27920i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$	−18082.1	−	1044.61i	−1.71408	−	0.0990235i
$482$		0			0
$483$		0			0
$484$	10648.0			1.00000
$485$		0			0
$486$		0			0
$487$	7940.26	+	7940.26i	0.738824	+	0.738824i	0.972351	−	0.233526i	$-0.0750265\pi$
$487$	7940.26	+	7940.26i	0.738824	+	0.738824i	−0.233526	+	0.972351i	$0.575026\pi$
$488$		0			0
$489$	−15070.5	+	15070.5i	−1.39368	+	1.39368i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−4867.69	−	4867.69i	−0.440657	−	0.440657i
$497$		0			0
$498$		0			0
$499$	−15751.9	−	15751.9i	−1.41313	−	1.41313i	−0.734195	−	0.678938i	$-0.762440\pi$
$499$	−15751.9	−	15751.9i	−1.41313	−	1.41313i	−0.678938	−	0.734195i	$-0.737560\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1314.63	+	11340.0i	−0.115157	+	0.993347i
$508$	−3040.00			−0.265508
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$			8741.03i			0.756713i
$512$		0			0
$513$	−14863.3	−	14863.3i	−1.27920	−	1.27920i
$514$		0			0
$515$		0			0
$516$	−9072.00			−0.773978
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	12040.0			1.00664			0.503320	−	0.864100i	$-0.332112\pi$
$523$	12040.0			1.00664			0.503320	+	0.864100i	$0.332112\pi$
$524$		0			0
$525$	3629.81	+	3629.81i	0.301748	+	0.301748i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$		0			0
$532$		−	9472.86i		−	0.771994i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5883.04	−	5883.04i	0.467526	−	0.467526i	−0.433586	−	0.901112i	$-0.642752\pi$
$541$	5883.04	−	5883.04i	0.467526	−	0.467526i	0.901112	+	0.433586i	$0.142752\pi$
$542$		0			0
$543$		−	17820.0i		−	1.40834i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1640.00			0.128193			0.0640963	−	0.997944i	$-0.479584\pi$
$547$	1640.00			0.128193			0.0640963	+	0.997944i	$0.479584\pi$
$548$		0			0
$549$	−25253.3			−1.96318
$550$		0			0
$551$		0			0
$552$		0			0
$553$	6098.08	−	6098.08i	0.468927	−	0.468927i
$554$		0			0
$555$		0			0
$556$			20608.0i			1.57190i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$	7638.34	−	6804.00i	0.577938	−	0.514810i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4073.99	−	4073.99i	0.301748	−	0.301748i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			23312.0i			1.70854i	0.519829	+	0.854270i	$0.325996\pi$
$571$			23312.0i			1.70854i	−0.519829	+	0.854270i	$0.674004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	13824.0i		−	1.00000i
$577$	1807.50	+	1807.50i	0.130411	+	0.130411i	0.769300	−	0.638888i	$-0.220605\pi$
$577$	1807.50	+	1807.50i	0.130411	+	0.130411i	−0.638888	+	0.769300i	$0.720605\pi$
$578$		0			0
$579$	−10620.2	+	10620.2i	−0.762281	+	0.762281i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	−11661.8			−0.817896
$589$		−	16115.5i		−	1.12738i
$590$		0			0
$591$		0			0
$592$	−17487.3	+	17487.3i	−1.21406	+	1.21406i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	27207.1i		−	1.86518i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	3117.69			0.211603			0.105801	−	0.994387i	$-0.466259\pi$
$601$	3117.69			0.211603			0.105801	+	0.994387i	$0.466259\pi$
$602$		0			0
$603$	20718.7	+	20718.7i	1.39922	+	1.39922i
$604$	20086.3	+	20086.3i	1.35315	+	1.35315i
$605$		0			0
$606$		0			0
$607$	28420.0			1.90038			0.950191	−	0.311667i	$-0.100887\pi$
$607$	28420.0			1.90038			0.950191	+	0.311667i	$0.100887\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−3744.59	−	3744.59i	−0.246725	−	0.246725i	0.572900	−	0.819625i	$-0.305818\pi$
$613$	−3744.59	−	3744.59i	−0.246725	−	0.246725i	−0.819625	+	0.572900i	$0.805818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	21044.3	−	21044.3i	1.36646	−	1.36646i	0.501040	−	0.865424i	$-0.332951\pi$
$619$	21044.3	−	21044.3i	1.36646	−	1.36646i	0.865424	−	0.501040i	$-0.167049\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	10368.0	+	11639.4i	0.665148	+	0.746712i
$625$	−15625.0			−1.00000
$626$		0			0
$627$		0			0
$628$			30800.0i			1.95709i
$629$		0			0
$630$		0			0
$631$	14876.3	−	14876.3i	0.938535	−	0.938535i	−0.0596825	−	0.998217i	$-0.519009\pi$
$631$	14876.3	−	14876.3i	0.938535	−	0.938535i	0.998217	+	0.0596825i	$0.0190088\pi$
$632$		0			0
$633$	5670.00			0.356023
$634$		0			0
$635$		0			0
$636$		0			0
$637$	9818.84	−	8746.32i	0.610733	−	0.544022i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	8338.15	+	8338.15i	0.511391	+	0.511391i	0.914953	−	0.403561i	$-0.132228\pi$
$643$	8338.15	+	8338.15i	0.511391	+	0.511391i	−0.403561	+	0.914953i	$0.632228\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	4417.20			0.265935
$652$	−23202.5	−	23202.5i	−1.39368	−	1.39368i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−21115.7	+	21115.7i	−1.25388	+	1.25388i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−23803.0	+	23803.0i	−1.40065	+	1.40065i	−0.602615	+	0.798032i	$0.705875\pi$
$661$	−23803.0	+	23803.0i	−1.40065	+	1.40065i	−0.798032	+	0.602615i	$0.794125\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−6781.19	−	6781.19i	−0.391893	−	0.391893i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	25315.7i		−	1.45000i	−0.688751	−	0.724998i	$-0.741841\pi$
$673$		−	25315.7i		−	1.45000i	0.688751	−	0.724998i	$-0.258159\pi$
$674$		0			0
$675$			17537.0i			1.00000i
$676$	−17459.1	−	2024.00i	−0.993347	−	0.115157i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−15098.8			−0.853371
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$	22883.5	−	22883.5i	1.27920	−	1.27920i
$685$		0			0
$686$		0			0
$687$	2166.99	−	2166.99i	0.120343	−	0.120343i
$688$		−	13967.3i		−	0.773978i
$689$		0			0
$690$		0			0
$691$	−24325.9	−	24325.9i	−1.33922	−	1.33922i	−0.896814	−	0.442408i	$-0.854124\pi$
$691$	−24325.9	−	24325.9i	−1.33922	−	1.33922i	−0.442408	−	0.896814i	$-0.645876\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−5588.46	+	5588.46i	−0.301748	+	0.301748i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−57895.0			−3.10605
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12617.0	−	12617.0i	0.668326	−	0.668326i	−0.289003	−	0.957328i	$-0.593324\pi$
$709$	12617.0	−	12617.0i	0.668326	−	0.668326i	0.957328	+	0.289003i	$0.0933236\pi$
$710$		0			0
$711$	29462.2			1.55403
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	5749.62	+	5749.62i	0.296986	+	0.296986i
$722$		0			0
$723$	−15928.6	+	15928.6i	−0.819352	+	0.819352i
$724$	27435.7			1.40834
$725$		0			0
$726$		0			0
$727$			10780.0i			0.549942i	0.961452	+	0.274971i	$0.0886683\pi$
$727$			10780.0i			0.549942i	−0.961452	+	0.274971i	$0.911332\pi$
$728$		0			0
$729$	19683.0			1.00000
$730$		0			0
$731$		0			0
$732$		−	38880.0i		−	1.96318i
$733$	25888.2	+	25888.2i	1.30451	+	1.30451i	0.925321	+	0.379184i	$0.123795\pi$
$733$	25888.2	+	25888.2i	1.30451	+	1.30451i	0.379184	+	0.925321i	$0.376205\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−28236.7	+	28236.7i	−1.40555	+	1.40555i	−0.624644	+	0.780910i	$0.714756\pi$
$739$	−28236.7	+	28236.7i	−1.40555	+	1.40555i	−0.780910	+	0.624644i	$0.785244\pi$
$740$		0			0
$741$	−2104.58	+	36429.9i	−0.104337	+	1.80605i
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	33827.0i		−	1.64363i	−0.569757	−	0.821813i	$-0.692963\pi$
$751$		−	33827.0i		−	1.64363i	0.569757	−	0.821813i	$-0.307037\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	6272.31	+	6272.31i	0.301748	+	0.301748i
$757$	3928.29			0.188608			0.0943039	−	0.995543i	$-0.469937\pi$
$757$	3928.29			0.188608			0.0943039	+	0.995543i	$0.469937\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$	−8586.02			−0.407385
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	21283.4			1.00000
$769$	18897.3	+	18897.3i	0.886156	+	0.886156i	0.994151	−	0.107995i	$-0.0344431\pi$
$769$	18897.3	+	18897.3i	0.886156	+	0.886156i	−0.107995	+	0.994151i	$0.534443\pi$
$770$		0			0
$771$		0			0
$772$	−16350.9	−	16350.9i	−0.762281	−	0.762281i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$	−9507.21	+	9507.21i	−0.440657	+	0.440657i
$776$		0			0
$777$		−	15868.9i		−	0.732679i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	17954.5i		−	0.817896i
$785$		0			0
$786$		0			0
$787$	25768.6	−	25768.6i	1.16715	−	1.16715i	0.184281	−	0.982874i	$-0.441004\pi$
$787$	25768.6	−	25768.6i	1.16715	−	1.16715i	0.982874	−	0.184281i	$-0.0589958\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	29160.0	+	32735.8i	1.30580	+	1.46593i
$794$		0			0
$795$		0			0
$796$	41888.0			1.86518
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−31898.5	+	31898.5i	−1.39922	+	1.39922i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−7602.95	−	7602.95i	−0.329193	−	0.329193i	0.523087	−	0.852280i	$-0.324780\pi$
$811$	−7602.95	−	7602.95i	−0.329193	−	0.329193i	−0.852280	+	0.523087i	$0.824780\pi$
$812$		0			0
$813$	−20975.4	+	20975.4i	−0.904844	+	0.904844i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	23120.7	−	23120.7i	0.990073	−	0.990073i
$818$		0			0
$819$	−9985.33	−	576.859i	−0.426026	−	0.0246118i
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$			45611.8i			1.93187i	0.258786	+	0.965935i	$0.416677\pi$
$823$			45611.8i			1.93187i	−0.258786	+	0.965935i	$0.583323\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		−	17066.0i		−	0.714990i	−0.933915	−	0.357495i	$-0.883631\pi$
$829$		−	17066.0i		−	0.714990i	0.933915	−	0.357495i	$-0.116369\pi$
$830$		0			0
$831$		−	43092.0i		−	1.79885i
$832$	−17920.0	+	15962.6i	−0.746712	+	0.665148i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	10670.6	+	10670.6i	0.440657	+	0.440657i
$838$		0			0
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$			8729.54i			0.356023i
$845$		0			0
$846$		0			0
$847$	−7438.24	−	7438.24i	−0.301748	−	0.301748i
$848$		0			0
$849$		−	29098.5i		−	1.17627i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−32043.4	+	32043.4i	−1.28622	+	1.28622i	−0.349151	+	0.937066i	$0.613530\pi$
$853$	−32043.4	+	32043.4i	−1.28622	+	1.28622i	−0.937066	+	0.349151i	$0.886470\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−39438.8			−1.56651			−0.783256	−	0.621699i	$-0.786443\pi$
$859$	−39438.8			−1.56651			−0.783256	+	0.621699i	$0.786443\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−25528.7			−1.00000
$868$			6800.72i			0.265935i
$869$		0			0
$870$		0			0
$871$	2933.67	−	50781.4i	0.114126	−	1.97550i
$872$		0			0
$873$	−36474.1	−	36474.1i	−1.41404	−	1.41404i
$874$		0			0
$875$		0			0
$876$	−32509.7	−	32509.7i	−1.25388	−	1.25388i
$877$	18309.6	+	18309.6i	0.704985	+	0.704985i	0.965476	−	0.260491i	$-0.0838846\pi$
$877$	18309.6	+	18309.6i	0.704985	+	0.704985i	−0.260491	+	0.965476i	$0.583885\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		−	20680.0i		−	0.788151i	−0.919078	−	0.394076i	$-0.871065\pi$
$883$		−	20680.0i		−	0.788151i	0.919078	−	0.394076i	$-0.128935\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	2123.61	+	2123.61i	0.0801167	+	0.0801167i
$890$		0			0
$891$		0			0
$892$	10440.3	−	10440.3i	0.391893	−	0.391893i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−27000.0			−1.00000
$901$		0			0
$902$		0			0
$903$	6337.31	+	6337.31i	0.233547	+	0.233547i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			44840.0i			1.64155i	0.571250	+	0.820776i	$0.306459\pi$
$907$			44840.0i			1.64155i	−0.571250	+	0.820776i	$0.693541\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	35231.5	+	35231.5i	1.27920	+	1.27920i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	3336.30	+	3336.30i	0.120343	+	0.120343i
$917$		0			0
$918$		0			0
$919$	−55650.8			−1.99755			−0.998776	−	0.0494625i	$-0.984249\pi$
$919$	−55650.8			−1.99755			−0.998776	+	0.0494625i	$0.984249\pi$
$920$		0			0
$921$	23512.5	−	23512.5i	0.841221	−	0.841221i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	34154.8	+	34154.8i	1.21406	+	1.21406i
$926$		0			0
$927$			27778.6i			0.984218i
$928$		0			0
$929$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$929$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$	29720.9	−	29720.9i	1.04625	−	1.04625i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	14466.1			0.504361			0.252181	−	0.967680i	$-0.418852\pi$
$937$	14466.1			0.504361			0.252181	+	0.967680i	$0.418852\pi$
$938$		0			0
$939$	24624.0			0.855776
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$			45360.0i			1.55403i
$949$	51754.4	+	2989.89i	1.77030	+	0.102272i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	18221.4i		−	0.611643i
$962$		0			0
$963$		0			0
$964$	−24523.7	−	24523.7i	−0.819352	−	0.819352i
$965$		0			0
$966$		0			0
$967$	8314.76	+	8314.76i	0.276510	+	0.276510i	0.831714	−	0.555204i	$-0.187360\pi$
$967$	8314.76	+	8314.76i	0.276510	+	0.276510i	−0.555204	+	0.831714i	$0.687360\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$			30304.0i			1.00000i
$973$	14395.9	−	14395.9i	0.474317	−	0.474317i
$974$		0			0
$975$	22733.2	−	20250.0i	0.746712	−	0.665148i
$976$	59859.7			1.96318
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−20741.2	−	20741.2i	−0.675041	−	0.675041i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−56087.5	−	3240.21i	−1.80605	−	0.104337i
$989$		0			0
$990$		0			0
$991$	−45628.0			−1.46258			−0.731292	−	0.682064i	$-0.761082\pi$
$991$	−45628.0			−1.46258			−0.731292	+	0.682064i	$0.761082\pi$
$992$		0			0
$993$	28607.7	+	28607.7i	0.914238	+	0.914238i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−28910.0			−0.918344			−0.459172	−	0.888347i	$-0.651854\pi$
$997$	−28910.0			−0.918344			−0.459172	+	0.888347i	$0.651854\pi$
$998$		0			0
$999$	38334.3	−	38334.3i	1.21406	−	1.21406i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.f.a.8.2	yes	4
3.2	odd	2	CM	39.4.f.a.8.2	yes	4
13.5	odd	4	inner	39.4.f.a.5.2	✓	4
39.5	even	4	inner	39.4.f.a.5.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.f.a.5.2	✓	4	13.5	odd	4	inner
39.4.f.a.5.2	✓	4	39.5	even	4	inner
39.4.f.a.8.2	yes	4	1.1	even	1	trivial
39.4.f.a.8.2	yes	4	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} +8.00000i q^{4} +(5.58846 - 5.58846i) q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} +8.00000i q^{4} +(5.58846 - 5.58846i) q^{7} +27.0000 q^{9} +41.5692i q^{12} +(-31.1769 - 35.0000i) q^{13} -64.0000 q^{16} +(-105.942 - 105.942i) q^{19} +(29.0385 - 29.0385i) q^{21} +125.000i q^{25} +140.296 q^{27} +(44.7077 + 44.7077i) q^{28} +(76.0577 + 76.0577i) q^{31} +216.000i q^{36} +(273.238 - 273.238i) q^{37} +(-162.000 - 181.865i) q^{39} +218.238i q^{43} -332.554 q^{48} +280.538i q^{49} +(280.000 - 249.415i) q^{52} +(-550.492 - 550.492i) q^{57} -935.307 q^{61} +(150.888 - 150.888i) q^{63} -512.000i q^{64} +(767.358 + 767.358i) q^{67} +(-782.061 + 782.061i) q^{73} +649.519i q^{75} +(847.538 - 847.538i) q^{76} +1091.19 q^{79} +729.000 q^{81} +(232.308 + 232.308i) q^{84} +(-369.827 - 21.3651i) q^{91} +(395.207 + 395.207i) q^{93} +(-1350.89 - 1350.89i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 40 q^{7} + 108 q^{9}+O(q^{10}) \) 4 * q - 40 * q^7 + 108 * q^9	\( 4 q - 40 q^{7} + 108 q^{9} - 256 q^{16} - 112 q^{19} + 324 q^{21} - 320 q^{28} + 616 q^{31} + 220 q^{37} - 648 q^{39} + 1120 q^{52} - 1620 q^{57} - 1080 q^{63} + 1760 q^{67} - 2380 q^{73} + 896 q^{76} + 2916 q^{81} + 2592 q^{84} - 544 q^{91} - 1620 q^{93} - 2660 q^{97}+O(q^{100}) \) 4 * q - 40 * q^7 + 108 * q^9 - 256 * q^16 - 112 * q^19 + 324 * q^21 - 320 * q^28 + 616 * q^31 + 220 * q^37 - 648 * q^39 + 1120 * q^52 - 1620 * q^57 - 1080 * q^63 + 1760 * q^67 - 2380 * q^73 + 896 * q^76 + 2916 * q^81 + 2592 * q^84 - 544 * q^91 - 1620 * q^93 - 2660 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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