Properties

Label 2475.2.a.u
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
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] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 2) q^{4} - q^{7} + (\beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 2) q^{4} - q^{7} + (\beta + 4) q^{8} + q^{11} + ( - \beta + 3) q^{13} - \beta q^{14} + 3 \beta q^{16} + ( - \beta + 4) q^{17} + 3 q^{19} + \beta q^{22} + (\beta + 4) q^{23} + (2 \beta - 4) q^{26} + ( - \beta - 2) q^{28} - 2 \beta q^{29} + (3 \beta - 3) q^{31} + (\beta + 4) q^{32} + (3 \beta - 4) q^{34} + (5 \beta - 2) q^{37} + 3 \beta q^{38} + 3 \beta q^{41} + ( - 3 \beta + 3) q^{43} + (\beta + 2) q^{44} + (5 \beta + 4) q^{46} + ( - \beta + 8) q^{47} - 6 q^{49} + 2 q^{52} + ( - 2 \beta + 4) q^{53} + ( - \beta - 4) q^{56} + ( - 2 \beta - 8) q^{58} + ( - \beta - 8) q^{59} + ( - \beta - 5) q^{61} + 12 q^{62} + ( - \beta + 4) q^{64} + (3 \beta - 1) q^{67} + (\beta + 4) q^{68} + ( - 5 \beta + 4) q^{71} + ( - 4 \beta - 6) q^{73} + (3 \beta + 20) q^{74} + (3 \beta + 6) q^{76} - q^{77} + ( - 3 \beta - 8) q^{79} + (3 \beta + 12) q^{82} - 12 q^{86} + (\beta + 4) q^{88} + (2 \beta + 12) q^{89} + (\beta - 3) q^{91} + (7 \beta + 12) q^{92} + (7 \beta - 4) q^{94} + ( - 2 \beta + 3) q^{97} - 6 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} + 2 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} + 7 q^{17} + 6 q^{19} + q^{22} + 9 q^{23} - 6 q^{26} - 5 q^{28} - 2 q^{29} - 3 q^{31} + 9 q^{32} - 5 q^{34} + q^{37} + 3 q^{38} + 3 q^{41} + 3 q^{43} + 5 q^{44} + 13 q^{46} + 15 q^{47} - 12 q^{49} + 4 q^{52} + 6 q^{53} - 9 q^{56} - 18 q^{58} - 17 q^{59} - 11 q^{61} + 24 q^{62} + 7 q^{64} + q^{67} + 9 q^{68} + 3 q^{71} - 16 q^{73} + 43 q^{74} + 15 q^{76} - 2 q^{77} - 19 q^{79} + 27 q^{82} - 24 q^{86} + 9 q^{88} + 26 q^{89} - 5 q^{91} + 31 q^{92} - q^{94} + 4 q^{97} - 6 q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 0 0.438447 0 0 −1.00000 2.43845 0 0
1.2 2.56155 0 4.56155 0 0 −1.00000 6.56155 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 2475.2.a.u yes 2
3.b odd 2 1 2475.2.a.p 2
5.b even 2 1 2475.2.a.q yes 2
5.c odd 4 2 2475.2.c.j 4
15.d odd 2 1 2475.2.a.v yes 2
15.e even 4 2 2475.2.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.p 2 3.b odd 2 1
2475.2.a.q yes 2 5.b even 2 1
2475.2.a.u yes 2 1.a even 1 1 trivial
2475.2.a.v yes 2 15.d odd 2 1
2475.2.c.i 4 15.e even 4 2
2475.2.c.j 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{29}^{2} + 2T_{29} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$19$ \( (T - 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$37$ \( T^{2} - T - 106 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$47$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 17T + 68 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$67$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 104 \) Copy content Toggle raw display
$73$ \( T^{2} + 16T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 19T + 52 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 26T + 152 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 13 \) Copy content Toggle raw display
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