Properties

Label 2475.2.a.l
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
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] = [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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + ( - \beta - 1) q^{7} + (\beta - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + ( - \beta - 1) q^{7} + (\beta - 3) q^{8} + q^{11} + 2 \beta q^{13} - q^{14} + 3 q^{16} + (\beta - 1) q^{17} + ( - \beta + 5) q^{19} + (\beta - 1) q^{22} - q^{23} + ( - 2 \beta + 4) q^{26} + (\beta + 3) q^{28} + ( - 2 \beta - 4) q^{29} + 6 \beta q^{31} + (\beta + 3) q^{32} + ( - 2 \beta + 3) q^{34} + ( - 2 \beta - 3) q^{37} + (6 \beta - 7) q^{38} + ( - 7 \beta + 1) q^{41} + (4 \beta - 6) q^{43} + ( - 2 \beta + 1) q^{44} + ( - \beta + 1) q^{46} + ( - 6 \beta - 1) q^{47} + (2 \beta - 4) q^{49} + (2 \beta - 8) q^{52} + (4 \beta - 2) q^{53} + (2 \beta + 1) q^{56} - 2 \beta q^{58} - 11 q^{59} + ( - 2 \beta + 6) q^{61} + ( - 6 \beta + 12) q^{62} + (2 \beta - 7) q^{64} + ( - 4 \beta - 6) q^{67} + (3 \beta - 5) q^{68} + (2 \beta - 5) q^{71} + (2 \beta - 6) q^{73} + ( - \beta - 1) q^{74} + ( - 11 \beta + 9) q^{76} + ( - \beta - 1) q^{77} + ( - 3 \beta + 9) q^{79} + (8 \beta - 15) q^{82} + ( - 6 \beta - 4) q^{83} + ( - 10 \beta + 14) q^{86} + (\beta - 3) q^{88} + (4 \beta + 2) q^{89} + ( - 2 \beta - 4) q^{91} + (2 \beta - 1) q^{92} + (5 \beta - 11) q^{94} + ( - 2 \beta + 3) q^{97} + ( - 6 \beta + 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{14} + 6 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{22} - 2 q^{23} + 8 q^{26} + 6 q^{28} - 8 q^{29} + 6 q^{32} + 6 q^{34} - 6 q^{37} - 14 q^{38} + 2 q^{41} - 12 q^{43} + 2 q^{44} + 2 q^{46} - 2 q^{47} - 8 q^{49} - 16 q^{52} - 4 q^{53} + 2 q^{56} - 22 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} - 12 q^{67} - 10 q^{68} - 10 q^{71} - 12 q^{73} - 2 q^{74} + 18 q^{76} - 2 q^{77} + 18 q^{79} - 30 q^{82} - 8 q^{83} + 28 q^{86} - 6 q^{88} + 4 q^{89} - 8 q^{91} - 2 q^{92} - 22 q^{94} + 6 q^{97} + 16 q^{98}+O(q^{100}) \) 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
−1.41421
1.41421
−2.41421 0 3.82843 0 0 0.414214 −4.41421 0 0
1.2 0.414214 0 −1.82843 0 0 −2.41421 −1.58579 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.l 2
3.b odd 2 1 825.2.a.f yes 2
5.b even 2 1 2475.2.a.w 2
5.c odd 4 2 2475.2.c.o 4
15.d odd 2 1 825.2.a.d 2
15.e even 4 2 825.2.c.d 4
33.d even 2 1 9075.2.a.w 2
165.d even 2 1 9075.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.d 2 15.d odd 2 1
825.2.a.f yes 2 3.b odd 2 1
825.2.c.d 4 15.e even 4 2
2475.2.a.l 2 1.a even 1 1 trivial
2475.2.a.w 2 5.b even 2 1
2475.2.c.o 4 5.c odd 4 2
9075.2.a.w 2 33.d even 2 1
9075.2.a.ca 2 165.d even 2 1

Hecke kernels

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

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 1 \) Copy content Toggle raw display
\( T_{29}^{2} + 8T_{29} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} - 10T + 23 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 72 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 97 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 71 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$59$ \( (T + 11)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 17 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$79$ \( T^{2} - 18T + 63 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
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