Properties

Label 1.34.a.a
Level $1$
Weight $34$
Character orbit 1.a
Self dual yes
Analytic conductor $6.898$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,34,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, 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("1.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(6.89828288810\)
Analytic rank: \(1\)
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: \( x^{2} - x - 589050 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(\beta = 72\sqrt{2356201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 60840) q^{2} + (312 \beta + 18959940) q^{3} + (121680 \beta + 7326116992) q^{4} + ( - 1004000 \beta - 90530768250) q^{5} + ( - 37942020 \beta - 4964461096608) q^{6} + (801594864 \beta - 33576540033400) q^{7} + ( - 6139193600 \beta - 14\!\cdots\!20) q^{8}+ \cdots + (11831002560 \beta - 40\!\cdots\!27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 60840) q^{2} + (312 \beta + 18959940) q^{3} + (121680 \beta + 7326116992) q^{4} + ( - 1004000 \beta - 90530768250) q^{5} + ( - 37942020 \beta - 4964461096608) q^{6} + (801594864 \beta - 33576540033400) q^{7} + ( - 6139193600 \beta - 14\!\cdots\!20) q^{8}+ \cdots + (61\!\cdots\!20 \beta - 46\!\cdots\!64) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 121680 q^{2} + 37919880 q^{3} + 14652233984 q^{4} - 181061536500 q^{5} - 9928922193216 q^{6} - 67153080066800 q^{7} - 28\!\cdots\!40 q^{8}+ \cdots - 80\!\cdots\!54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 121680 q^{2} + 37919880 q^{3} + 14652233984 q^{4} - 181061536500 q^{5} - 9928922193216 q^{6} - 67153080066800 q^{7} - 28\!\cdots\!40 q^{8}+ \cdots - 92\!\cdots\!28 q^{99}+O(q^{100}) \) 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
767.996
−766.996
−171359. 5.34420e7 2.07741e10 −2.01492e11 −9.15779e12 5.50153e13 −2.08788e15 −2.70301e15 3.45276e16
1.2 49679.4 −1.55221e7 −6.12189e9 2.04307e10 −7.71130e11 −1.22168e14 −7.30875e14 −5.31812e15 1.01499e15
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.34.a.a 2
3.b odd 2 1 9.34.a.b 2
4.b odd 2 1 16.34.a.b 2
5.b even 2 1 25.34.a.a 2
5.c odd 4 2 25.34.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 1.a even 1 1 trivial
9.34.a.b 2 3.b odd 2 1
16.34.a.b 2 4.b odd 2 1
25.34.a.a 2 5.b even 2 1
25.34.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{34}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 121680 T - 8513040384 \) Copy content Toggle raw display
$3$ \( T^{2} + \cdots - 829533439462896 \) Copy content Toggle raw display
$5$ \( T^{2} + 181061536500 T - 41\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 67153080066800 T - 67\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 84\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 35\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 35\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 33\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 41\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 26\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 42\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 50\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 31\!\cdots\!04 \) Copy content Toggle raw display
show more
show less