Properties

Label 2.34.a.b
Level $2$
Weight $34$
Character orbit 2.a
Self dual yes
Analytic conductor $13.797$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2,34,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

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: yes
Analytic conductor: \(13.7965657762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 19957422 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 10560\sqrt{79829689}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 65536 q^{2} + ( - \beta + 4178244) q^{3} + 4294967296 q^{4} + ( - 3996 \beta - 2666238330) q^{5} + ( - 65536 \beta + 273825398784) q^{6} + (896238 \beta + 66359547937928) q^{7} + 281474976710656 q^{8} + ( - 8356488 \beta + 33\!\cdots\!13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 65536 q^{2} + ( - \beta + 4178244) q^{3} + 4294967296 q^{4} + ( - 3996 \beta - 2666238330) q^{5} + ( - 65536 \beta + 273825398784) q^{6} + (896238 \beta + 66359547937928) q^{7} + 281474976710656 q^{8} + ( - 8356488 \beta + 33\!\cdots\!13) q^{9}+ \cdots + (80\!\cdots\!45 \beta - 43\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 131072 q^{2} + 8356488 q^{3} + 8589934592 q^{4} - 5332476660 q^{5} + 547650797568 q^{6} + 132719095875856 q^{7} + 562949953421312 q^{8} + 67\!\cdots\!26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 131072 q^{2} + 8356488 q^{3} + 8589934592 q^{4} - 5332476660 q^{5} + 547650797568 q^{6} + 132719095875856 q^{7} + 562949953421312 q^{8} + 67\!\cdots\!26 q^{9}+ \cdots - 86\!\cdots\!88 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4467.87
−4466.87
65536.0 −9.01727e7 4.29497e9 −3.79693e11 −5.90956e12 1.50920e14 2.81475e14 2.57205e15 −2.48835e16
1.2 65536.0 9.85292e7 4.29497e9 3.74360e11 6.45721e12 −1.82013e13 2.81475e14 4.14894e15 2.45341e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.34.a.b 2
3.b odd 2 1 18.34.a.e 2
4.b odd 2 1 16.34.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.34.a.b 2 1.a even 1 1 trivial
16.34.a.c 2 4.b odd 2 1
18.34.a.e 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8356488T_{3} - 8884638284346864 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 65536)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8356488 T - 88\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{2} + 5332476660 T - 14\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} - 132719095875856 T - 27\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 37\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 94\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 10\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 22\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 52\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 84\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 74\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 75\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 50\!\cdots\!36 \) Copy content Toggle raw display
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