Properties

Label 245.4.a.b
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
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] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(14.4554679514\)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - 6 q^{3} - 7 q^{4} + 5 q^{5} - 6 q^{6} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 6 q^{3} - 7 q^{4} + 5 q^{5} - 6 q^{6} - 15 q^{8} + 9 q^{9} + 5 q^{10} - 44 q^{11} + 42 q^{12} - 6 q^{13} - 30 q^{15} + 41 q^{16} + 24 q^{17} + 9 q^{18} + 114 q^{19} - 35 q^{20} - 44 q^{22} - 52 q^{23} + 90 q^{24} + 25 q^{25} - 6 q^{26} + 108 q^{27} + 146 q^{29} - 30 q^{30} + 276 q^{31} + 161 q^{32} + 264 q^{33} + 24 q^{34} - 63 q^{36} - 210 q^{37} + 114 q^{38} + 36 q^{39} - 75 q^{40} - 444 q^{41} + 492 q^{43} + 308 q^{44} + 45 q^{45} - 52 q^{46} + 612 q^{47} - 246 q^{48} + 25 q^{50} - 144 q^{51} + 42 q^{52} + 50 q^{53} + 108 q^{54} - 220 q^{55} - 684 q^{57} + 146 q^{58} - 294 q^{59} + 210 q^{60} - 450 q^{61} + 276 q^{62} - 167 q^{64} - 30 q^{65} + 264 q^{66} - 668 q^{67} - 168 q^{68} + 312 q^{69} - 308 q^{71} - 135 q^{72} - 12 q^{73} - 210 q^{74} - 150 q^{75} - 798 q^{76} + 36 q^{78} + 596 q^{79} + 205 q^{80} - 891 q^{81} - 444 q^{82} + 966 q^{83} + 120 q^{85} + 492 q^{86} - 876 q^{87} + 660 q^{88} + 408 q^{89} + 45 q^{90} + 364 q^{92} - 1656 q^{93} + 612 q^{94} + 570 q^{95} - 966 q^{96} + 1200 q^{97} - 396 q^{99}+O(q^{100}) \) 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 −6.00000 −7.00000 5.00000 −6.00000 0 −15.0000 9.00000 5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(7\) \( -1 \)

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 245.4.a.b 1
3.b odd 2 1 2205.4.a.k 1
5.b even 2 1 1225.4.a.g 1
7.b odd 2 1 245.4.a.c yes 1
7.c even 3 2 245.4.e.d 2
7.d odd 6 2 245.4.e.c 2
21.c even 2 1 2205.4.a.n 1
35.c odd 2 1 1225.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.b 1 1.a even 1 1 trivial
245.4.a.c yes 1 7.b odd 2 1
245.4.e.c 2 7.d odd 6 2
245.4.e.d 2 7.c even 3 2
1225.4.a.f 1 35.c odd 2 1
1225.4.a.g 1 5.b even 2 1
2205.4.a.k 1 3.b odd 2 1
2205.4.a.n 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{3} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 6 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 44 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 24 \) Copy content Toggle raw display
$19$ \( T - 114 \) Copy content Toggle raw display
$23$ \( T + 52 \) Copy content Toggle raw display
$29$ \( T - 146 \) Copy content Toggle raw display
$31$ \( T - 276 \) Copy content Toggle raw display
$37$ \( T + 210 \) Copy content Toggle raw display
$41$ \( T + 444 \) Copy content Toggle raw display
$43$ \( T - 492 \) Copy content Toggle raw display
$47$ \( T - 612 \) Copy content Toggle raw display
$53$ \( T - 50 \) Copy content Toggle raw display
$59$ \( T + 294 \) Copy content Toggle raw display
$61$ \( T + 450 \) Copy content Toggle raw display
$67$ \( T + 668 \) Copy content Toggle raw display
$71$ \( T + 308 \) Copy content Toggle raw display
$73$ \( T + 12 \) Copy content Toggle raw display
$79$ \( T - 596 \) Copy content Toggle raw display
$83$ \( T - 966 \) Copy content Toggle raw display
$89$ \( T - 408 \) Copy content Toggle raw display
$97$ \( T - 1200 \) Copy content Toggle raw display
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