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Properties

Label 1445.2.d.j
Level $1445$
Weight $2$
Character orbit 1445.d
Analytic conductor $11.538$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1445,2,Mod(866,1445)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1445, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1445.866");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.d (of order \(2\), degree \(1\), not 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: \(11.5383830921\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{2} + 24 q^{4} + 24 q^{8} - 24 q^{9} - 16 q^{13} + 16 q^{15} + 24 q^{16} + 8 q^{18} + 32 q^{21} - 24 q^{25} - 32 q^{26} - 16 q^{30} + 56 q^{32} - 32 q^{35} - 24 q^{36} - 48 q^{38} + 32 q^{43}+ \cdots - 120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
866.1 −2.35190 1.56935i 3.53144 − 1.00000i − 3.69096i − 3.58212i −3.60181 0.537139 2.35190i
866.2 −2.35190 − 1.56935i 3.53144 1.00000i 3.69096i 3.58212i −3.60181 0.537139 − 2.35190i
866.3 −2.04505 3.19566i 2.18224 − 1.00000i − 6.53528i 1.17743i −0.372688 −7.21221 2.04505i
866.4 −2.04505 − 3.19566i 2.18224 1.00000i 6.53528i − 1.17743i −0.372688 −7.21221 − 2.04505i
866.5 −1.43840 − 0.109907i 0.0689897 − 1.00000i 0.158090i − 0.695085i 2.77756 2.98792 1.43840i
866.6 −1.43840 0.109907i 0.0689897 1.00000i − 0.158090i 0.695085i 2.77756 2.98792 − 1.43840i
866.7 −0.747914 3.07503i −1.44062 − 1.00000i − 2.29986i − 3.23262i 2.57329 −6.45581 0.747914i
866.8 −0.747914 − 3.07503i −1.44062 1.00000i 2.29986i 3.23262i 2.57329 −6.45581 − 0.747914i
866.9 −0.360254 − 0.0542373i −1.87022 − 1.00000i 0.0195392i 0.298718i 1.39426 2.99706 0.360254i
866.10 −0.360254 0.0542373i −1.87022 1.00000i − 0.0195392i − 0.298718i 1.39426 2.99706 − 0.360254i
866.11 −0.301687 − 1.06101i −1.90899 − 1.00000i 0.320094i 2.50984i 1.17929 1.87425 0.301687i
866.12 −0.301687 1.06101i −1.90899 1.00000i − 0.320094i − 2.50984i 1.17929 1.87425 − 0.301687i
866.13 0.962871 − 2.64897i −1.07288 − 1.00000i − 2.55062i − 3.09463i −2.95879 −4.01706 − 0.962871i
866.14 0.962871 2.64897i −1.07288 1.00000i 2.55062i 3.09463i −2.95879 −4.01706 0.962871i
866.15 1.55041 1.14040i 0.403772 − 1.00000i 1.76809i 3.74146i −2.47481 1.69949 − 1.55041i
866.16 1.55041 − 1.14040i 0.403772 1.00000i − 1.76809i − 3.74146i −2.47481 1.69949 1.55041i
866.17 1.55555 − 3.00797i 0.419729 1.00000i − 4.67904i 3.45467i −2.45819 −6.04787 1.55555i
866.18 1.55555 3.00797i 0.419729 − 1.00000i 4.67904i − 3.45467i −2.45819 −6.04787 − 1.55555i
866.19 1.80583 0.687917i 1.26102 1.00000i 1.24226i 4.34193i −1.33447 2.52677 1.80583i
866.20 1.80583 − 0.687917i 1.26102 − 1.00000i − 1.24226i − 4.34193i −1.33447 2.52677 − 1.80583i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 866.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1445.2.d.j 24
17.b even 2 1 inner 1445.2.d.j 24
17.c even 4 1 1445.2.a.p 12
17.c even 4 1 1445.2.a.q 12
17.e odd 16 2 85.2.l.a ✓ 24
51.i even 16 2 765.2.be.b 24
85.j even 4 1 7225.2.a.bq 12
85.j even 4 1 7225.2.a.bs 12
85.o even 16 2 425.2.n.c 24
85.p odd 16 2 425.2.m.b 24
85.r even 16 2 425.2.n.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.l.a ✓ 24 17.e odd 16 2
425.2.m.b 24 85.p odd 16 2
425.2.n.c 24 85.o even 16 2
425.2.n.f 24 85.r even 16 2
765.2.be.b 24 51.i even 16 2
1445.2.a.p 12 17.c even 4 1
1445.2.a.q 12 17.c even 4 1
1445.2.d.j 24 1.a even 1 1 trivial
1445.2.d.j 24 17.b even 2 1 inner
7225.2.a.bq 12 85.j even 4 1
7225.2.a.bs 12 85.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1445, [\chi])\):

\( T_{2}^{12} - 4 T_{2}^{11} - 10 T_{2}^{10} + 52 T_{2}^{9} + 21 T_{2}^{8} - 232 T_{2}^{7} + 44 T_{2}^{6} + \cdots + 17 \) Copy content Toggle raw display
\( T_{3}^{24} + 48 T_{3}^{22} + 972 T_{3}^{20} + 10832 T_{3}^{18} + 72824 T_{3}^{16} + 305592 T_{3}^{14} + \cdots + 4 \) Copy content Toggle raw display