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Properties

Label 1445.2.d.i
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 / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,2,Mod(866,1445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-6,0,42,0,0,0,-24,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5383830921\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 42 q^{4} - 24 q^{8} - 42 q^{9} + 18 q^{13} - 6 q^{15} + 78 q^{16} - 18 q^{18} - 54 q^{19} + 12 q^{21} - 24 q^{25} - 12 q^{26} - 18 q^{30} - 24 q^{32} + 12 q^{35} - 96 q^{36} - 6 q^{38}+ \cdots - 84 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
866.1 −2.77092 0.449467i 5.67798 1.00000i − 1.24543i − 1.06235i −10.1914 2.79798 − 2.77092i
866.2 −2.77092 − 0.449467i 5.67798 − 1.00000i 1.24543i 1.06235i −10.1914 2.79798 2.77092i
866.3 −2.69854 2.66942i 5.28213 − 1.00000i − 7.20354i 0.656507i −8.85696 −4.12579 2.69854i
866.4 −2.69854 − 2.66942i 5.28213 1.00000i 7.20354i − 0.656507i −8.85696 −4.12579 − 2.69854i
866.5 −2.27524 − 1.70271i 3.17670 1.00000i 3.87408i 2.70724i −2.67727 0.100765 − 2.27524i
866.6 −2.27524 1.70271i 3.17670 − 1.00000i − 3.87408i − 2.70724i −2.67727 0.100765 2.27524i
866.7 −1.99107 2.94118i 1.96434 1.00000i − 5.85608i − 3.47942i 0.0709948 −5.65053 − 1.99107i
866.8 −1.99107 − 2.94118i 1.96434 − 1.00000i 5.85608i 3.47942i 0.0709948 −5.65053 1.99107i
866.9 −1.66011 − 2.23148i 0.755971 − 1.00000i 3.70450i − 1.12916i 2.06523 −1.97948 1.66011i
866.10 −1.66011 2.23148i 0.755971 1.00000i − 3.70450i 1.12916i 2.06523 −1.97948 − 1.66011i
866.11 −0.341036 − 1.39350i −1.88369 1.00000i 0.475233i 3.70869i 1.32448 1.05816 − 0.341036i
866.12 −0.341036 1.39350i −1.88369 − 1.00000i − 0.475233i − 3.70869i 1.32448 1.05816 0.341036i
866.13 0.0540929 0.605946i −1.99707 1.00000i 0.0327774i 4.08179i −0.216213 2.63283 0.0540929i
866.14 0.0540929 − 0.605946i −1.99707 − 1.00000i − 0.0327774i − 4.08179i −0.216213 2.63283 − 0.0540929i
866.15 0.704111 3.11994i −1.50423 − 1.00000i 2.19679i 4.06346i −2.46736 −6.73405 − 0.704111i
866.16 0.704111 − 3.11994i −1.50423 1.00000i − 2.19679i − 4.06346i −2.46736 −6.73405 0.704111i
866.17 1.02921 1.23410i −0.940729 1.00000i 1.27015i − 0.979236i −3.02662 1.47699 1.02921i
866.18 1.02921 − 1.23410i −0.940729 − 1.00000i − 1.27015i 0.979236i −3.02662 1.47699 − 1.02921i
866.19 1.82030 − 2.83144i 1.31351 − 1.00000i − 5.15409i 3.11716i −1.24963 −5.01706 − 1.82030i
866.20 1.82030 2.83144i 1.31351 1.00000i 5.15409i − 3.11716i −1.24963 −5.01706 1.82030i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 866.24
Significant digits:
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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.i 24
17.b even 2 1 inner 1445.2.d.i 24
17.c even 4 1 1445.2.a.r ✓ 12
17.c even 4 1 1445.2.a.s yes 12
85.j even 4 1 7225.2.a.bn 12
85.j even 4 1 7225.2.a.bo 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1445.2.a.r ✓ 12 17.c even 4 1
1445.2.a.s yes 12 17.c even 4 1
1445.2.d.i 24 1.a even 1 1 trivial
1445.2.d.i 24 17.b even 2 1 inner
7225.2.a.bn 12 85.j even 4 1
7225.2.a.bo 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 \(S_{2}^{\mathrm{new}}(1445, [\chi])\):

\( T_{2}^{12} + 3 T_{2}^{11} - 18 T_{2}^{10} - 55 T_{2}^{9} + 114 T_{2}^{8} + 354 T_{2}^{7} - 309 T_{2}^{6} + \cdots + 9 \) Copy content Toggle raw display
\( T_{3}^{24} + 57 T_{3}^{22} + 1416 T_{3}^{20} + 20129 T_{3}^{18} + 180849 T_{3}^{16} + 1071366 T_{3}^{14} + \cdots + 310249 \) Copy content Toggle raw display