Properties

Label 5175.2.a.bf
Level $5175$
Weight $2$
Character orbit 5175.a
Self dual yes
Analytic conductor $41.323$
Analytic rank $1$
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] = [5175,2,Mod(1,5175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5175, 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("5175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5175.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: \(41.3225830460\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1725)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 1) q^{4} + (\beta - 1) q^{7} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 1) q^{4} + (\beta - 1) q^{7} - 3 q^{8} + ( - \beta - 1) q^{11} + ( - \beta + 3) q^{13} - 3 q^{14} + (\beta - 2) q^{16} - q^{17} - 2 \beta q^{19} + (2 \beta + 3) q^{22} + q^{23} + ( - 2 \beta + 3) q^{26} + (\beta + 2) q^{28} + ( - 2 \beta + 4) q^{29} + (3 \beta - 3) q^{31} + (\beta + 3) q^{32} + \beta q^{34} + (2 \beta + 5) q^{37} + (2 \beta + 6) q^{38} + (2 \beta - 3) q^{41} + ( - 2 \beta + 1) q^{43} + ( - 3 \beta - 4) q^{44} - \beta q^{46} + (2 \beta - 3) q^{47} + ( - \beta - 3) q^{49} + \beta q^{52} + (5 \beta - 3) q^{53} + ( - 3 \beta + 3) q^{56} + ( - 2 \beta + 6) q^{58} + (\beta - 3) q^{59} + (3 \beta - 11) q^{61} - 9 q^{62} + ( - 6 \beta + 1) q^{64} + ( - \beta + 10) q^{67} + ( - \beta - 1) q^{68} + (3 \beta - 7) q^{71} + ( - 4 \beta - 3) q^{73} + ( - 7 \beta - 6) q^{74} + ( - 4 \beta - 6) q^{76} + ( - \beta - 2) q^{77} + ( - 2 \beta + 5) q^{79} + (\beta - 6) q^{82} + (4 \beta - 7) q^{83} + (\beta + 6) q^{86} + (3 \beta + 3) q^{88} + ( - 2 \beta - 7) q^{89} + (3 \beta - 6) q^{91} + (\beta + 1) q^{92} + (\beta - 6) q^{94} + (\beta - 3) q^{97} + (4 \beta + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} - q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} - q^{7} - 6 q^{8} - 3 q^{11} + 5 q^{13} - 6 q^{14} - 3 q^{16} - 2 q^{17} - 2 q^{19} + 8 q^{22} + 2 q^{23} + 4 q^{26} + 5 q^{28} + 6 q^{29} - 3 q^{31} + 7 q^{32} + q^{34} + 12 q^{37} + 14 q^{38} - 4 q^{41} - 11 q^{44} - q^{46} - 4 q^{47} - 7 q^{49} + q^{52} - q^{53} + 3 q^{56} + 10 q^{58} - 5 q^{59} - 19 q^{61} - 18 q^{62} - 4 q^{64} + 19 q^{67} - 3 q^{68} - 11 q^{71} - 10 q^{73} - 19 q^{74} - 16 q^{76} - 5 q^{77} + 8 q^{79} - 11 q^{82} - 10 q^{83} + 13 q^{86} + 9 q^{88} - 16 q^{89} - 9 q^{91} + 3 q^{92} - 11 q^{94} - 5 q^{97} + 10 q^{98}+O(q^{100}) \) 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
2.30278
−1.30278
−2.30278 0 3.30278 0 0 1.30278 −3.00000 0 0
1.2 1.30278 0 −0.302776 0 0 −2.30278 −3.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 5175.2.a.bf 2
3.b odd 2 1 1725.2.a.bc yes 2
5.b even 2 1 5175.2.a.bm 2
15.d odd 2 1 1725.2.a.w 2
15.e even 4 2 1725.2.b.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1725.2.a.w 2 15.d odd 2 1
1725.2.a.bc yes 2 3.b odd 2 1
1725.2.b.r 4 15.e even 4 2
5175.2.a.bf 2 1.a even 1 1 trivial
5175.2.a.bm 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\):

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 23 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 13 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} + T - 81 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$61$ \( T^{2} + 19T + 61 \) Copy content Toggle raw display
$67$ \( T^{2} - 19T + 87 \) Copy content Toggle raw display
$71$ \( T^{2} + 11T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T - 27 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T - 27 \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 51 \) Copy content Toggle raw display
$97$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
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