Properties

Label 245.4.a.d
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: no (minimal twist has level 35)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + 8 q^{3} - 7 q^{4} + 5 q^{5} + 8 q^{6} - 15 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 8 q^{3} - 7 q^{4} + 5 q^{5} + 8 q^{6} - 15 q^{8} + 37 q^{9} + 5 q^{10} + 12 q^{11} - 56 q^{12} + 78 q^{13} + 40 q^{15} + 41 q^{16} + 94 q^{17} + 37 q^{18} - 40 q^{19} - 35 q^{20} + 12 q^{22} + 32 q^{23} - 120 q^{24} + 25 q^{25} + 78 q^{26} + 80 q^{27} - 50 q^{29} + 40 q^{30} + 248 q^{31} + 161 q^{32} + 96 q^{33} + 94 q^{34} - 259 q^{36} - 434 q^{37} - 40 q^{38} + 624 q^{39} - 75 q^{40} - 402 q^{41} - 68 q^{43} - 84 q^{44} + 185 q^{45} + 32 q^{46} - 536 q^{47} + 328 q^{48} + 25 q^{50} + 752 q^{51} - 546 q^{52} + 22 q^{53} + 80 q^{54} + 60 q^{55} - 320 q^{57} - 50 q^{58} + 560 q^{59} - 280 q^{60} + 278 q^{61} + 248 q^{62} - 167 q^{64} + 390 q^{65} + 96 q^{66} - 164 q^{67} - 658 q^{68} + 256 q^{69} + 672 q^{71} - 555 q^{72} - 82 q^{73} - 434 q^{74} + 200 q^{75} + 280 q^{76} + 624 q^{78} - 1000 q^{79} + 205 q^{80} - 359 q^{81} - 402 q^{82} + 448 q^{83} + 470 q^{85} - 68 q^{86} - 400 q^{87} - 180 q^{88} + 870 q^{89} + 185 q^{90} - 224 q^{92} + 1984 q^{93} - 536 q^{94} - 200 q^{95} + 1288 q^{96} - 1026 q^{97} + 444 q^{99}+O(q^{100}) \) 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 8.00000 −7.00000 5.00000 8.00000 0 −15.0000 37.0000 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.d 1
3.b odd 2 1 2205.4.a.i 1
5.b even 2 1 1225.4.a.e 1
7.b odd 2 1 35.4.a.a 1
7.c even 3 2 245.4.e.b 2
7.d odd 6 2 245.4.e.e 2
21.c even 2 1 315.4.a.c 1
28.d even 2 1 560.4.a.p 1
35.c odd 2 1 175.4.a.a 1
35.f even 4 2 175.4.b.a 2
56.e even 2 1 2240.4.a.b 1
56.h odd 2 1 2240.4.a.bk 1
105.g even 2 1 1575.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.a 1 7.b odd 2 1
175.4.a.a 1 35.c odd 2 1
175.4.b.a 2 35.f even 4 2
245.4.a.d 1 1.a even 1 1 trivial
245.4.e.b 2 7.c even 3 2
245.4.e.e 2 7.d odd 6 2
315.4.a.c 1 21.c even 2 1
560.4.a.p 1 28.d even 2 1
1225.4.a.e 1 5.b even 2 1
1575.4.a.g 1 105.g even 2 1
2205.4.a.i 1 3.b odd 2 1
2240.4.a.b 1 56.e even 2 1
2240.4.a.bk 1 56.h odd 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
\( T_{3} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 8 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T - 78 \) Copy content Toggle raw display
$17$ \( T - 94 \) Copy content Toggle raw display
$19$ \( T + 40 \) Copy content Toggle raw display
$23$ \( T - 32 \) Copy content Toggle raw display
$29$ \( T + 50 \) Copy content Toggle raw display
$31$ \( T - 248 \) Copy content Toggle raw display
$37$ \( T + 434 \) Copy content Toggle raw display
$41$ \( T + 402 \) Copy content Toggle raw display
$43$ \( T + 68 \) Copy content Toggle raw display
$47$ \( T + 536 \) Copy content Toggle raw display
$53$ \( T - 22 \) Copy content Toggle raw display
$59$ \( T - 560 \) Copy content Toggle raw display
$61$ \( T - 278 \) Copy content Toggle raw display
$67$ \( T + 164 \) Copy content Toggle raw display
$71$ \( T - 672 \) Copy content Toggle raw display
$73$ \( T + 82 \) Copy content Toggle raw display
$79$ \( T + 1000 \) Copy content Toggle raw display
$83$ \( T - 448 \) Copy content Toggle raw display
$89$ \( T - 870 \) Copy content Toggle raw display
$97$ \( T + 1026 \) Copy content Toggle raw display
show more
show less