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Properties

Label 296.1.k.a
Level $296$
Weight $1$
Character orbit 296.k
Analytic conductor $0.148$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [296,1,Mod(105,296)] 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("296.105"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 296 = 2^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 296.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.147723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.810448.1

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{3} + ( - i - 1) q^{5} - q^{7} - i q^{11} + (i + 1) q^{13} + (i - 1) q^{15} + i q^{21} + (i + 1) q^{23} + i q^{25} - i q^{27} + ( - i + 1) q^{29} - q^{33} + (i + 1) q^{35} - q^{37} + ( - i + 1) q^{39} + \cdots + ( - i - 1) q^{91} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{13} - 2 q^{15} + 2 q^{23} + 2 q^{29} - 2 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 2 q^{43} - 2 q^{47} - 2 q^{53} - 2 q^{55} + 2 q^{69} + 2 q^{71} + 2 q^{75} - 2 q^{81} - 2 q^{83}+ \cdots - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/296\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(-i\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
105.1
− 1.00000i
1.00000i
0 1.00000i 0 −1.00000 + 1.00000i 0 −1.00000 0 0 0
265.1 0 − 1.00000i 0 −1.00000 − 1.00000i 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 296.1.k.a ✓ 2
3.b odd 2 1 2664.1.y.a 2
4.b odd 2 1 592.1.k.a 2
8.b even 2 1 2368.1.k.c 2
8.d odd 2 1 2368.1.k.e 2
37.d odd 4 1 inner 296.1.k.a ✓ 2
111.g even 4 1 2664.1.y.a 2
148.g even 4 1 592.1.k.a 2
296.j even 4 1 2368.1.k.e 2
296.m odd 4 1 2368.1.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.1.k.a ✓ 2 1.a even 1 1 trivial
296.1.k.a ✓ 2 37.d odd 4 1 inner
592.1.k.a 2 4.b odd 2 1
592.1.k.a 2 148.g even 4 1
2368.1.k.c 2 8.b even 2 1
2368.1.k.c 2 296.m odd 4 1
2368.1.k.e 2 8.d odd 2 1
2368.1.k.e 2 296.j even 4 1
2664.1.y.a 2 3.b odd 2 1
2664.1.y.a 2 111.g even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(296, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$47$ \( (T + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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