Properties

Label 245.4.a.e
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} - 2 q^{3} + q^{4} + 5 q^{5} - 6 q^{6} - 21 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} - 2 q^{3} + q^{4} + 5 q^{5} - 6 q^{6} - 21 q^{8} - 23 q^{9} + 15 q^{10} - 45 q^{11} - 2 q^{12} + 59 q^{13} - 10 q^{15} - 71 q^{16} - 54 q^{17} - 69 q^{18} - 121 q^{19} + 5 q^{20} - 135 q^{22} + 69 q^{23} + 42 q^{24} + 25 q^{25} + 177 q^{26} + 100 q^{27} - 162 q^{29} - 30 q^{30} - 88 q^{31} - 45 q^{32} + 90 q^{33} - 162 q^{34} - 23 q^{36} - 259 q^{37} - 363 q^{38} - 118 q^{39} - 105 q^{40} + 195 q^{41} - 286 q^{43} - 45 q^{44} - 115 q^{45} + 207 q^{46} + 45 q^{47} + 142 q^{48} + 75 q^{50} + 108 q^{51} + 59 q^{52} + 597 q^{53} + 300 q^{54} - 225 q^{55} + 242 q^{57} - 486 q^{58} - 360 q^{59} - 10 q^{60} + 392 q^{61} - 264 q^{62} + 433 q^{64} + 295 q^{65} + 270 q^{66} - 280 q^{67} - 54 q^{68} - 138 q^{69} + 48 q^{71} + 483 q^{72} + 668 q^{73} - 777 q^{74} - 50 q^{75} - 121 q^{76} - 354 q^{78} + 782 q^{79} - 355 q^{80} + 421 q^{81} + 585 q^{82} + 768 q^{83} - 270 q^{85} - 858 q^{86} + 324 q^{87} + 945 q^{88} - 1194 q^{89} - 345 q^{90} + 69 q^{92} + 176 q^{93} + 135 q^{94} - 605 q^{95} + 90 q^{96} + 902 q^{97} + 1035 q^{99}+O(q^{100}) \) 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
3.00000 −2.00000 1.00000 5.00000 −6.00000 0 −21.0000 −23.0000 15.0000
\(n\): 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.e 1
3.b odd 2 1 2205.4.a.e 1
5.b even 2 1 1225.4.a.b 1
7.b odd 2 1 245.4.a.f 1
7.c even 3 2 35.4.e.a 2
7.d odd 6 2 245.4.e.a 2
21.c even 2 1 2205.4.a.g 1
21.h odd 6 2 315.4.j.b 2
28.g odd 6 2 560.4.q.b 2
35.c odd 2 1 1225.4.a.a 1
35.j even 6 2 175.4.e.b 2
35.l odd 12 4 175.4.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 7.c even 3 2
175.4.e.b 2 35.j even 6 2
175.4.k.b 4 35.l odd 12 4
245.4.a.e 1 1.a even 1 1 trivial
245.4.a.f 1 7.b odd 2 1
245.4.e.a 2 7.d odd 6 2
315.4.j.b 2 21.h odd 6 2
560.4.q.b 2 28.g odd 6 2
1225.4.a.a 1 35.c odd 2 1
1225.4.a.b 1 5.b even 2 1
2205.4.a.e 1 3.b odd 2 1
2205.4.a.g 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 45 \) Copy content Toggle raw display
$13$ \( T - 59 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T + 121 \) Copy content Toggle raw display
$23$ \( T - 69 \) Copy content Toggle raw display
$29$ \( T + 162 \) Copy content Toggle raw display
$31$ \( T + 88 \) Copy content Toggle raw display
$37$ \( T + 259 \) Copy content Toggle raw display
$41$ \( T - 195 \) Copy content Toggle raw display
$43$ \( T + 286 \) Copy content Toggle raw display
$47$ \( T - 45 \) Copy content Toggle raw display
$53$ \( T - 597 \) Copy content Toggle raw display
$59$ \( T + 360 \) Copy content Toggle raw display
$61$ \( T - 392 \) Copy content Toggle raw display
$67$ \( T + 280 \) Copy content Toggle raw display
$71$ \( T - 48 \) Copy content Toggle raw display
$73$ \( T - 668 \) Copy content Toggle raw display
$79$ \( T - 782 \) Copy content Toggle raw display
$83$ \( T - 768 \) Copy content Toggle raw display
$89$ \( T + 1194 \) Copy content Toggle raw display
$97$ \( T - 902 \) Copy content Toggle raw display
show more
show less