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Properties

Label 1445.2.b.i
Level $1445$
Weight $2$
Character orbit 1445.b
Analytic conductor $11.538$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1445,2,Mod(579,1445)]
 
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([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1445.579");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.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: 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} + 8 q^{9} + 8 q^{15} - 8 q^{16} + 16 q^{19} - 32 q^{21} + 8 q^{25} - 48 q^{26} - 56 q^{30} + 8 q^{35} + 8 q^{36} + 56 q^{49} - 16 q^{50} - 24 q^{55} + 64 q^{59} - 88 q^{60} + 104 q^{64} - 16 q^{66}+ \cdots + 128 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
579.1 − 2.30578i − 1.87767i −3.31660 1.99336 + 1.01316i −4.32949 − 1.54574i 3.03579i −0.525655 2.33613 − 4.59625i
579.2 − 2.30578i 1.87767i −3.31660 −1.99336 − 1.01316i 4.32949 1.54574i 3.03579i −0.525655 −2.33613 + 4.59625i
579.3 − 2.10495i − 1.80557i −2.43082 −0.681948 + 2.12954i −3.80063 − 2.68338i 0.906851i −0.260076 4.48258 + 1.43547i
579.4 − 2.10495i 1.80557i −2.43082 0.681948 − 2.12954i 3.80063 2.68338i 0.906851i −0.260076 −4.48258 − 1.43547i
579.5 − 1.74231i − 0.598698i −1.03565 2.14480 − 0.632313i −1.04312 − 1.03798i − 1.68020i 2.64156 −1.10169 − 3.73692i
579.6 − 1.74231i 0.598698i −1.03565 −2.14480 + 0.632313i 1.04312 1.03798i − 1.68020i 2.64156 1.10169 + 3.73692i
579.7 − 0.951739i − 2.47483i 1.09419 2.23233 + 0.129310i −2.35540 − 0.533377i − 2.94486i −3.12480 0.123069 − 2.12459i
579.8 − 0.951739i 2.47483i 1.09419 −2.23233 − 0.129310i 2.35540 0.533377i − 2.94486i −3.12480 −0.123069 + 2.12459i
579.9 − 0.499161i − 0.171281i 1.75084 1.06651 − 1.96534i −0.0854968 − 3.87301i − 1.87227i 2.97066 −0.981018 − 0.532362i
579.10 − 0.499161i 0.171281i 1.75084 −1.06651 + 1.96534i 0.0854968 3.87301i − 1.87227i 2.97066 0.981018 + 0.532362i
579.11 − 0.248918i − 1.64368i 1.93804 0.916806 − 2.03948i −0.409143 1.43109i − 0.980251i 0.298307 −0.507663 − 0.228210i
579.12 − 0.248918i 1.64368i 1.93804 −0.916806 + 2.03948i 0.409143 − 1.43109i − 0.980251i 0.298307 0.507663 + 0.228210i
579.13 0.248918i − 1.64368i 1.93804 −0.916806 − 2.03948i 0.409143 1.43109i 0.980251i 0.298307 0.507663 − 0.228210i
579.14 0.248918i 1.64368i 1.93804 0.916806 + 2.03948i −0.409143 − 1.43109i 0.980251i 0.298307 −0.507663 + 0.228210i
579.15 0.499161i − 0.171281i 1.75084 −1.06651 − 1.96534i 0.0854968 − 3.87301i 1.87227i 2.97066 0.981018 − 0.532362i
579.16 0.499161i 0.171281i 1.75084 1.06651 + 1.96534i −0.0854968 3.87301i 1.87227i 2.97066 −0.981018 + 0.532362i
579.17 0.951739i − 2.47483i 1.09419 −2.23233 + 0.129310i 2.35540 − 0.533377i 2.94486i −3.12480 −0.123069 − 2.12459i
579.18 0.951739i 2.47483i 1.09419 2.23233 − 0.129310i −2.35540 0.533377i 2.94486i −3.12480 0.123069 + 2.12459i
579.19 1.74231i − 0.598698i −1.03565 −2.14480 − 0.632313i 1.04312 − 1.03798i 1.68020i 2.64156 1.10169 − 3.73692i
579.20 1.74231i 0.598698i −1.03565 2.14480 + 0.632313i −1.04312 1.03798i 1.68020i 2.64156 −1.10169 + 3.73692i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 579.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c 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.b.i 24
5.b even 2 1 inner 1445.2.b.i 24
5.c odd 4 2 7225.2.a.by 24
17.b even 2 1 inner 1445.2.b.i 24
17.e odd 16 2 85.2.m.a ✓ 24
51.i even 16 2 765.2.bh.b 24
85.c even 2 1 inner 1445.2.b.i 24
85.g odd 4 2 7225.2.a.by 24
85.o even 16 2 425.2.m.e 24
85.p odd 16 2 85.2.m.a ✓ 24
85.r even 16 2 425.2.m.e 24
255.be even 16 2 765.2.bh.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.m.a ✓ 24 17.e odd 16 2
85.2.m.a ✓ 24 85.p odd 16 2
425.2.m.e 24 85.o even 16 2
425.2.m.e 24 85.r even 16 2
765.2.bh.b 24 51.i even 16 2
765.2.bh.b 24 255.be even 16 2
1445.2.b.i 24 1.a even 1 1 trivial
1445.2.b.i 24 5.b even 2 1 inner
1445.2.b.i 24 17.b even 2 1 inner
1445.2.b.i 24 85.c even 2 1 inner
7225.2.a.by 24 5.c odd 4 2
7225.2.a.by 24 85.g odd 4 2

Hecke kernels

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

\( T_{2}^{12} + 14T_{2}^{10} + 69T_{2}^{8} + 140T_{2}^{6} + 103T_{2}^{4} + 22T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{12} - 60T_{11}^{10} + 1270T_{11}^{8} - 11252T_{11}^{6} + 38776T_{11}^{4} - 27936T_{11}^{2} + 162 \) Copy content Toggle raw display