LMFDB - Newform orbit 16.9.c.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [16,9,Mod(15,16)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("16.15"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	16.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,516] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.51805776098$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of $\beta = 8\sqrt{-3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta q^{3} + 258 q^{5} - 238 \beta q^{7} + 6369 q^{9} - 1671 \beta q^{11} + 19138 q^{13} - 258 \beta q^{15} - 58686 q^{17} + 11011 \beta q^{19} - 45696 q^{21} + 19494 \beta q^{23} - 324061 q^{25} + \cdots - 10642599 \beta q^{99} +O(q^{100}) $ q - b * q^3 + 258 * q^5 - 238b q^7 + 6369 * q^9 - 1671b q^11 + 19138 * q^13 - 258b q^15 - 58686 * q^17 + 11011b q^19 - 45696 * q^21 + 19494b q^23 - 324061 * q^25 - 12930b q^27 + 842178 * q^29 - 75864b q^31 - 320832 * q^33 - 61404b q^35 + 2548610 * q^37 - 19138b q^39 - 4324158 * q^41 + 147009b q^43 + 1643202 * q^45 + 522012b q^47 - 5110847 * q^49 + 58686b q^51 + 1192194 * q^53 - 431118b q^55 + 2114112 * q^57 - 24411b q^59 + 8414786 * q^61 - 1515822b q^63 + 4937604 * q^65 - 1257521b q^67 + 3742848 * q^69 + 2228322b q^71 + 12735874 * q^73 + 324061b q^75 - 76358016 * q^77 + 453700b q^79 + 39304449 * q^81 + 5995347b q^83 - 15140988 * q^85 - 842178b q^87 - 16802814 * q^89 - 4554844b q^91 - 14565888 * q^93 + 2840838b q^95 + 120994882 * q^97 - 10642599b q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 516 q^{5} + 12738 q^{9} + 38276 q^{13} - 117372 q^{17} - 91392 q^{21} - 648122 q^{25} + 1684356 q^{29} - 641664 q^{33} + 5097220 q^{37} - 8648316 q^{41} + 3286404 q^{45} - 10221694 q^{49} + 2384388 q^{53}+ \cdots + 241989764 q^{97}+O(q^{100}) $ 2 * q + 516 * q^5 + 12738 * q^9 + 38276 * q^13 - 117372 * q^17 - 91392 * q^21 - 648122 * q^25 + 1684356 * q^29 - 641664 * q^33 + 5097220 * q^37 - 8648316 * q^41 + 3286404 * q^45 - 10221694 * q^49 + 2384388 * q^53 + 4228224 * q^57 + 16829572 * q^61 + 9875208 * q^65 + 7485696 * q^69 + 25471748 * q^73 - 152716032 * q^77 + 78608898 * q^81 - 30281976 * q^85 - 33605628 * q^89 - 29131776 * q^93 + 241989764 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$.

$n$	$5$	$15$
$\chi(n)$	$1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

15.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−

13.8564i

258.000

−

3297.82i

6369.00

15.2

13.8564i

258.000

3297.82i

6369.00

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	16.9.c.b	✓	2
3.b	odd	2	1		144.9.g.d		2
4.b	odd	2	1	inner	16.9.c.b	✓	2
5.b	even	2	1		400.9.b.e		2
5.c	odd	4	2		400.9.h.a		4
8.b	even	2	1		64.9.c.c		2
8.d	odd	2	1		64.9.c.c		2
12.b	even	2	1		144.9.g.d		2
16.e	even	4	2		256.9.d.d		4
16.f	odd	4	2		256.9.d.d		4
20.d	odd	2	1		400.9.b.e		2
20.e	even	4	2		400.9.h.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
16.9.c.b	✓	2	1.a	even	1	1	trivial
16.9.c.b	✓	2	4.b	odd	2	1	inner
64.9.c.c		2	8.b	even	2	1
64.9.c.c		2	8.d	odd	2	1
144.9.g.d		2	3.b	odd	2	1
144.9.g.d		2	12.b	even	2	1
256.9.d.d		4	16.e	even	4	2
256.9.d.d		4	16.f	odd	4	2
400.9.b.e		2	5.b	even	2	1
400.9.b.e		2	20.d	odd	2	1
400.9.h.a		4	5.c	odd	4	2
400.9.h.a		4	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} + 192 $ T3^2 + 192 acting on $S_{9}^{\mathrm{new}}(16, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 192 $ T^2 + 192
$5$	$ (T - 258)^{2} $ (T - 258)^2
$7$	$ T^{2} + 10875648 $ T^2 + 10875648
$11$	$ T^{2} + 536110272 $ T^2 + 536110272
$13$	$ (T - 19138)^{2} $ (T - 19138)^2
$17$	$ (T + 58686)^{2} $ (T + 58686)^2
$19$	$ T^{2} + 23278487232 $ T^2 + 23278487232
$23$	$ T^{2} + 72963078912 $ T^2 + 72963078912
$29$	$ (T - 842178)^{2} $ (T - 842178)^2
$31$	$ T^{2} + 1105026527232 $ T^2 + 1105026527232
$37$	$ (T - 2548610)^{2} $ (T - 2548610)^2
$41$	$ (T + 4324158)^{2} $ (T + 4324158)^2
$43$	$ T^{2} + 4149436047552 $ T^2 + 4149436047552
$47$	$ T^{2} + 52319333403648 $ T^2 + 52319333403648
$53$	$ (T - 1192194)^{2} $ (T - 1192194)^2
$59$	$ T^{2} + 114412208832 $ T^2 + 114412208832
$61$	$ (T - 8414786)^{2} $ (T - 8414786)^2
$67$	$ T^{2} + 303620940564672 $ T^2 + 303620940564672
$71$	$ T^{2} + 953360435651328 $ T^2 + 953360435651328
$73$	$ (T - 12735874)^{2} $ (T - 12735874)^2
$79$	$ T^{2} + 39521988480000 $ T^2 + 39521988480000
$83$	$ T^{2} + 69\!\cdots\!28 $ T^2 + 6901283644878528
$89$	$ (T + 16802814)^{2} $ (T + 16802814)^2
$97$	$ (T - 120994882)^{2} $ (T - 120994882)^2
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Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta q^{3} + 258 q^{5} - 238 \beta q^{7} + 6369 q^{9} - 1671 \beta q^{11} + 19138 q^{13} - 258 \beta q^{15} - 58686 q^{17} + 11011 \beta q^{19} - 45696 q^{21} + 19494 \beta q^{23} - 324061 q^{25} + \cdots - 10642599 \beta q^{99} +O(q^{100}) \) q - b * q^3 + 258 * q^5 - 238b q^7 + 6369 * q^9 - 1671b q^11 + 19138 * q^13 - 258b q^15 - 58686 * q^17 + 11011b q^19 - 45696 * q^21 + 19494b q^23 - 324061 * q^25 - 12930b q^27 + 842178 * q^29 - 75864b q^31 - 320832 * q^33 - 61404b q^35 + 2548610 * q^37 - 19138b q^39 - 4324158 * q^41 + 147009b q^43 + 1643202 * q^45 + 522012b q^47 - 5110847 * q^49 + 58686b q^51 + 1192194 * q^53 - 431118b q^55 + 2114112 * q^57 - 24411b q^59 + 8414786 * q^61 - 1515822b q^63 + 4937604 * q^65 - 1257521b q^67 + 3742848 * q^69 + 2228322b q^71 + 12735874 * q^73 + 324061b q^75 - 76358016 * q^77 + 453700b q^79 + 39304449 * q^81 + 5995347b q^83 - 15140988 * q^85 - 842178b q^87 - 16802814 * q^89 - 4554844b q^91 - 14565888 * q^93 + 2840838b q^95 + 120994882 * q^97 - 10642599b q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 516 q^{5} + 12738 q^{9} + 38276 q^{13} - 117372 q^{17} - 91392 q^{21} - 648122 q^{25} + 1684356 q^{29} - 641664 q^{33} + 5097220 q^{37} - 8648316 q^{41} + 3286404 q^{45} - 10221694 q^{49} + 2384388 q^{53}+ \cdots + 241989764 q^{97}+O(q^{100}) \) 2 * q + 516 * q^5 + 12738 * q^9 + 38276 * q^13 - 117372 * q^17 - 91392 * q^21 - 648122 * q^25 + 1684356 * q^29 - 641664 * q^33 + 5097220 * q^37 - 8648316 * q^41 + 3286404 * q^45 - 10221694 * q^49 + 2384388 * q^53 + 4228224 * q^57 + 16829572 * q^61 + 9875208 * q^65 + 7485696 * q^69 + 25471748 * q^73 - 152716032 * q^77 + 78608898 * q^81 - 30281976 * q^85 - 33605628 * q^89 - 29131776 * q^93 + 241989764 * q^97

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