Properties

Label 34.2.a.a
Level $34$
Weight $2$
Character orbit 34.a
Self dual yes
Analytic conductor $0.271$
Analytic rank $0$
Dimension $1$
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] = [34,2,Mod(1,34)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(34, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("34.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 34 = 2 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 34.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: \(0.271491366872\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 4 q^{14} + q^{16} - q^{17} + q^{18} - 4 q^{19} + 8 q^{21} + 6 q^{22} - 2 q^{24} - 5 q^{25} + 2 q^{26} + 4 q^{27} - 4 q^{28} - 4 q^{31} + q^{32} - 12 q^{33} - q^{34} + q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} + 6 q^{41} + 8 q^{42} + 8 q^{43} + 6 q^{44} - 2 q^{48} + 9 q^{49} - 5 q^{50} + 2 q^{51} + 2 q^{52} - 6 q^{53} + 4 q^{54} - 4 q^{56} + 8 q^{57} - 4 q^{61} - 4 q^{62} - 4 q^{63} + q^{64} - 12 q^{66} + 8 q^{67} - q^{68} + q^{72} + 2 q^{73} - 4 q^{74} + 10 q^{75} - 4 q^{76} - 24 q^{77} - 4 q^{78} + 8 q^{79} - 11 q^{81} + 6 q^{82} + 8 q^{84} + 8 q^{86} + 6 q^{88} - 6 q^{89} - 8 q^{91} + 8 q^{93} - 2 q^{96} + 14 q^{97} + 9 q^{98} + 6 q^{99}+O(q^{100}) \) 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
0
1.00000 −2.00000 1.00000 0 −2.00000 −4.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(17\) \( +1 \)

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 34.2.a.a 1
3.b odd 2 1 306.2.a.a 1
4.b odd 2 1 272.2.a.d 1
5.b even 2 1 850.2.a.e 1
5.c odd 4 2 850.2.c.b 2
7.b odd 2 1 1666.2.a.m 1
8.b even 2 1 1088.2.a.l 1
8.d odd 2 1 1088.2.a.d 1
11.b odd 2 1 4114.2.a.a 1
12.b even 2 1 2448.2.a.k 1
13.b even 2 1 5746.2.a.b 1
15.d odd 2 1 7650.2.a.ci 1
17.b even 2 1 578.2.a.a 1
17.c even 4 2 578.2.b.a 2
17.d even 8 4 578.2.c.e 4
17.e odd 16 8 578.2.d.e 8
20.d odd 2 1 6800.2.a.b 1
24.f even 2 1 9792.2.a.bj 1
24.h odd 2 1 9792.2.a.y 1
51.c odd 2 1 5202.2.a.d 1
68.d odd 2 1 4624.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 1.a even 1 1 trivial
272.2.a.d 1 4.b odd 2 1
306.2.a.a 1 3.b odd 2 1
578.2.a.a 1 17.b even 2 1
578.2.b.a 2 17.c even 4 2
578.2.c.e 4 17.d even 8 4
578.2.d.e 8 17.e odd 16 8
850.2.a.e 1 5.b even 2 1
850.2.c.b 2 5.c odd 4 2
1088.2.a.d 1 8.d odd 2 1
1088.2.a.l 1 8.b even 2 1
1666.2.a.m 1 7.b odd 2 1
2448.2.a.k 1 12.b even 2 1
4114.2.a.a 1 11.b odd 2 1
4624.2.a.a 1 68.d odd 2 1
5202.2.a.d 1 51.c odd 2 1
5746.2.a.b 1 13.b even 2 1
6800.2.a.b 1 20.d odd 2 1
7650.2.a.ci 1 15.d odd 2 1
9792.2.a.y 1 24.h odd 2 1
9792.2.a.bj 1 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(34))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 4 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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