LMFDB - Newform orbit 4.41.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,41,Mod(3,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 41, names="a")

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

chi := DirichletCharacter("4.3");

S:= CuspForms(chi, 41);

N := Newforms(S);

Level:	$N$	$=$	$4 = 2^{2}$
Weight:	$k$	$=$	$41$
Character orbit:	$[\chi]$	$=$	4.b (of order $2$ , degree $1$ , 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:	yes
Analytic conductor:	$40.5369140786$
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 + 1048576 q^{2} + 1099511627776 q^{4} + 182008936336226 q^{5} + 11\!\cdots\!76 q^{8} + 12\!\cdots\!01 q^{9} + 19\!\cdots\!76 q^{10} - 15\!\cdots\!98 q^{13} + 12\!\cdots\!76 q^{16} - 75\!\cdots\!98 q^{17}+ \cdots + 66\!\cdots\!76 q^{98}+O(q^{100})

q + 1048576 * q^2 + 1099511627776 * q^4 + 182008936336226 * q^5 + 1152921504606846976 * q^8 + 12157665459056928801 * q^9 + 190850202427694514176 * q^10 - 1589900269312604162398 * q^13 + 1208925819614629174706176 * q^16 - 7565475025188088957756798 * q^17 + 12748236216396078174437376 * q^18 + 200120941860822202896613376 * q^20 + 24032305888515086764969532451 * q^25 - 1667131264794733222190645248 * q^26 - 312717248594471233529540479198 * q^29 + 1267650600228229401496703205376 * q^32 - 7932975540011625566968792219648 * q^34 + 13367494538843734067838845976576 * q^36 + 43857017456118466671167150613602 * q^37 + 209842016732653502224519267352576 * q^40 - 101691394325222881712409889310398 * q^41 + 2212803758534626401134640679045026 * q^45 + 6366805760909027985741435139224001 * q^49 + 25199699179355595619664692459339776 * q^50 - 1748113833113402183191778031566848 * q^52 + 8318642270094614106091300338712802 * q^53 - 327907801662196268169471437515522048 * q^58 - 900441575672965352599394973899589598 * q^61 + 1329227995784915872903807060280344576 * q^64 - 289376056898266342937130835634429948 * q^65 - 8318327759843230290509868270509621248 * q^68 + 14016833953562607293918185758734155776 * q^72 - 7832507478698914491237343863157425598 * q^73 + 45987415936066877308185766121808330752 * q^74 + 220035302537458878748577515283494731776 * q^80 + 147808829414345923316083210206383297601 * q^81 - 106631155495964908414471912093539893248 * q^82 - 1376984062212766676893554148657665164348 * q^85 - 1824813187152655735552249267753788705598 * q^89 + 2320292913909204413196156984670317182976 * q^90 - 5682706651209893817497323457510706259198 * q^97 + 6676079717550944929176811092546946072576 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4\mathbb{Z}\right)^\times$ .

$n$	$3$
$\chi(n)$	$-1$

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}

3.1

1.04858e6

1.09951e12

1.82009e14

1.15292e18

1.21577e19

1.90850e20

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4.41.b.a	✓	1
4.b	odd	2	1	CM	4.41.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.41.b.a	✓	1	1.a	even	1	1	trivial
4.41.b.a	✓	1	4.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}$ T3 acting on $S_{41}^{\mathrm{new}}(4, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 1048576$ T - 1048576
$3$	$T$ T
$5$	$T - 182008936336226$ T - 182008936336226
$7$	$T$ T
$11$	$T$ T
$13$	$T + 15\!\cdots\!98$ T + 1589900269312604162398
$17$	$T + 75\!\cdots\!98$ T + 7565475025188088957756798
$19$	$T$ T
$23$	$T$ T
$29$	$T + 31\!\cdots\!98$ T + 312717248594471233529540479198
$31$	$T$ T
$37$	$T - 43\!\cdots\!02$ T - 43857017456118466671167150613602
$41$	$T + 10\!\cdots\!98$ T + 101691394325222881712409889310398
$43$	$T$ T
$47$	$T$ T
$53$	$T - 83\!\cdots\!02$ T - 8318642270094614106091300338712802
$59$	$T$ T
$61$	$T + 90\!\cdots\!98$ T + 900441575672965352599394973899589598
$67$	$T$ T
$71$	$T$ T
$73$	$T + 78\!\cdots\!98$ T + 7832507478698914491237343863157425598
$79$	$T$ T
$83$	$T$ T
$89$	$T + 18\!\cdots\!98$ T + 1824813187152655735552249267753788705598
$97$	$T + 56\!\cdots\!98$ T + 5682706651209893817497323457510706259198
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