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Properties

Label 954.2.c.d
Level $954$
Weight $2$
Character orbit 954.c
Analytic conductor $7.618$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [954,2,Mod(847,954)] 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("954.847"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(954, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 954 = 2 \cdot 3^{2} \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 954.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,0,2,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.61772835283\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 318)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + 4 i q^{5} + q^{7} + i q^{8} + 4 q^{10} - 5 q^{11} + 2 q^{13} - i q^{14} + q^{16} - i q^{19} - 4 i q^{20} + 5 i q^{22} + 3 i q^{23} - 11 q^{25} - 2 i q^{26} - q^{28} - 5 q^{29} + \cdots + 6 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{7} + 8 q^{10} - 10 q^{11} + 4 q^{13} + 2 q^{16} - 22 q^{25} - 2 q^{28} - 10 q^{29} - 20 q^{37} - 2 q^{38} - 8 q^{40} + 12 q^{43} + 10 q^{44} + 6 q^{46} - 12 q^{47} - 12 q^{49} - 4 q^{52}+ \cdots + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/954\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(425\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
847.1
1.00000i
− 1.00000i
− 1.00000i 0 −1.00000 4.00000i 0 1.00000 1.00000i 0 4.00000
847.2 1.00000i 0 −1.00000 − 4.00000i 0 1.00000 − 1.00000i 0 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 954.2.c.d 2
3.b odd 2 1 318.2.c.a ✓ 2
12.b even 2 1 2544.2.i.a 2
53.b even 2 1 inner 954.2.c.d 2
159.d odd 2 1 318.2.c.a ✓ 2
636.g even 2 1 2544.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
318.2.c.a ✓ 2 3.b odd 2 1
318.2.c.a ✓ 2 159.d odd 2 1
954.2.c.d 2 1.a even 1 1 trivial
954.2.c.d 2 53.b even 2 1 inner
2544.2.i.a 2 12.b even 2 1
2544.2.i.a 2 636.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(954, [\chi])\):

\( T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 5)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 36 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9 \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 53 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 169 \) Copy content Toggle raw display
$71$ \( T^{2} + 49 \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 256 \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( (T - 5)^{2} \) Copy content Toggle raw display
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