Properties

Label 275.2.a.a
Level $275$
Weight $2$
Character orbit 275.a
Self dual yes
Analytic conductor $2.196$
Analytic rank $1$
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] = [275,2,Mod(1,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(2.19588605559\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} - q^{11} - 2 q^{13} - q^{16} - 6 q^{17} + 3 q^{18} - 4 q^{19} + q^{22} - 4 q^{23} + 2 q^{26} + 6 q^{29} - 8 q^{31} - 5 q^{32} + 6 q^{34} + 3 q^{36} + 2 q^{37} + 4 q^{38} + 2 q^{41} - 4 q^{43} + q^{44} + 4 q^{46} + 12 q^{47} - 7 q^{49} + 2 q^{52} + 2 q^{53} - 6 q^{58} + 4 q^{59} - 10 q^{61} + 8 q^{62} + 7 q^{64} + 16 q^{67} + 6 q^{68} + 8 q^{71} - 9 q^{72} - 14 q^{73} - 2 q^{74} + 4 q^{76} + 8 q^{79} + 9 q^{81} - 2 q^{82} + 4 q^{83} + 4 q^{86} - 3 q^{88} + 10 q^{89} + 4 q^{92} - 12 q^{94} - 10 q^{97} + 7 q^{98} + 3 q^{99}+O(q^{100}) \) 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 0 −1.00000 0 0 0 3.00000 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(11\) \( +1 \)

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 275.2.a.a 1
3.b odd 2 1 2475.2.a.i 1
4.b odd 2 1 4400.2.a.p 1
5.b even 2 1 55.2.a.a 1
5.c odd 4 2 275.2.b.b 2
11.b odd 2 1 3025.2.a.f 1
15.d odd 2 1 495.2.a.a 1
15.e even 4 2 2475.2.c.f 2
20.d odd 2 1 880.2.a.h 1
20.e even 4 2 4400.2.b.n 2
35.c odd 2 1 2695.2.a.c 1
40.e odd 2 1 3520.2.a.n 1
40.f even 2 1 3520.2.a.p 1
55.d odd 2 1 605.2.a.b 1
55.h odd 10 4 605.2.g.c 4
55.j even 10 4 605.2.g.a 4
60.h even 2 1 7920.2.a.i 1
65.d even 2 1 9295.2.a.b 1
165.d even 2 1 5445.2.a.i 1
220.g even 2 1 9680.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 5.b even 2 1
275.2.a.a 1 1.a even 1 1 trivial
275.2.b.b 2 5.c odd 4 2
495.2.a.a 1 15.d odd 2 1
605.2.a.b 1 55.d odd 2 1
605.2.g.a 4 55.j even 10 4
605.2.g.c 4 55.h odd 10 4
880.2.a.h 1 20.d odd 2 1
2475.2.a.i 1 3.b odd 2 1
2475.2.c.f 2 15.e even 4 2
2695.2.a.c 1 35.c odd 2 1
3025.2.a.f 1 11.b odd 2 1
3520.2.a.n 1 40.e odd 2 1
3520.2.a.p 1 40.f even 2 1
4400.2.a.p 1 4.b odd 2 1
4400.2.b.n 2 20.e even 4 2
5445.2.a.i 1 165.d even 2 1
7920.2.a.i 1 60.h even 2 1
9295.2.a.b 1 65.d even 2 1
9680.2.a.r 1 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(275))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 16 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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