LMFDB - Newform orbit 15.25.d.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [15,25,Mod(14,15)] 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("15.14"); S:= CuspForms(chi, 25); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 25, names="a")

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 25 $
Character orbit:	$[\chi]$	$=$	15.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-8143] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.7450728387$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 8143 q^{2} + 531441 q^{3} + 49531233 q^{4} + 244140625 q^{5} - 4327524063 q^{6} - 266715960431 q^{8} + 282429536481 q^{9} - 1988037109375 q^{10} + 26322927996753 q^{12} + 129746337890625 q^{15} + 13\!\cdots\!05 q^{16}+ \cdots - 15\!\cdots\!43 q^{98}+O(q^{100}) $

q - 8143 * q^2 + 531441 * q^3 + 49531233 * q^4 + 244140625 * q^5 - 4327524063 * q^6 - 266715960431 * q^8 + 282429536481 * q^9 - 1988037109375 * q^10 + 26322927996753 * q^12 + 129746337890625 * q^15 + 1340871871002305 * q^16 + 432524144062082 * q^17 - 2299823715564783 * q^18 + 1904424373265762 * q^19 + 12092586181640625 * q^20 - 37214344554868798 * q^23 - 141743796727411071 * q^24 + 59604644775390625 * q^25 + 150094635296999121 * q^27 - 1056524429443359375 * q^30 + 1458725602323329282 * q^31 - 6443968366773429519 * q^32 - 3522044105097533726 * q^34 + 13989083177522411073 * q^36 - 15507727671503099966 * q^38 - 65116201277099609375 * q^40 + 68952523554931640625 * q^45 + 303036407710296622114 * q^46 - 51418335280297668478 * q^47 + 712594287997335971505 * q^48 + 191581231380566414401 * q^49 - 485360622406005859375 * q^50 + 229861063644496920162 * q^51 + 347328853598147231522 * q^53 - 1222220615223463842303 * q^54 + 1012089193352729823042 * q^57 + 6426496092957275390625 * q^60 - 5299065689612439877918 * q^61 - 11878402579718870343326 * q^62 + 29977137402506229090337 * q^64 + 21423454157664550007106 * q^68 - 19777228484584028877918 * q^69 - 75328465076612066983311 * q^72 + 31676352024078369140625 * q^75 + 94328487363105428544546 * q^76 - 20553733200969854686078 * q^79 + 327361296631422119140625 * q^80 + 79766443076872509863361 * q^81 - 25938127244873567233438 * q^83 + 105596714858906738281250 * q^85 - 561480399307808349609375 * q^90 - 1843272371089487718167934 * q^92 + 775226592824312436955362 * q^93 + 418699504187463914416354 * q^94 + 464947356754336425781250 * q^95 - 3424588992806438157006879 * q^96 - 1560045967131952312467343 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$-1$

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} $

14.1

−8143.00

531441.

4.95312e7

2.44141e8

−4.32752e9

−2.66716e11

2.82430e11

−1.98804e12

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	15.25.d.a	✓	1
3.b	odd	2	1		15.25.d.b	yes	1
5.b	even	2	1		15.25.d.b	yes	1
15.d	odd	2	1	CM	15.25.d.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
15.25.d.a	✓	1	1.a	even	1	1	trivial
15.25.d.a	✓	1	15.d	odd	2	1	CM
15.25.d.b	yes	1	3.b	odd	2	1
15.25.d.b	yes	1	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} + 8143 $ T2 + 8143 acting on $S_{25}^{\mathrm{new}}(15, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 8143 $ T + 8143
$3$	$ T - 531441 $ T - 531441
$5$	$ T - 244140625 $ T - 244140625
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T - 432524144062082 $ T - 432524144062082
$19$	$ T - 1904424373265762 $ T - 1904424373265762
$23$	$ T + 37\!\cdots\!98 $ T + 37214344554868798
$29$	$ T $ T
$31$	$ T - 14\!\cdots\!82 $ T - 1458725602323329282
$37$	$ T $ T
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T + 51\!\cdots\!78 $ T + 51418335280297668478
$53$	$ T - 34\!\cdots\!22 $ T - 347328853598147231522
$59$	$ T $ T
$61$	$ T + 52\!\cdots\!18 $ T + 5299065689612439877918
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T + 20\!\cdots\!78 $ T + 20553733200969854686078
$83$	$ T + 25\!\cdots\!38 $ T + 25938127244873567233438
$89$	$ T $ T
$97$	$ T $ T
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 25 \)
Character orbit:	\([\chi]\)	\(=\)	15.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(-1\)	\(-1\)

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