Properties

Label 2475.2.a.o
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{13}) \)
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 275)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 1) q^{4} + (\beta + 2) q^{7} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 1) q^{4} + (\beta + 2) q^{7} - 3 q^{8} + q^{11} + 5 q^{13} + ( - 3 \beta - 3) q^{14} + (\beta - 2) q^{16} + (3 \beta - 3) q^{17} - q^{19} - \beta q^{22} + ( - \beta + 6) q^{23} - 5 \beta q^{26} + (4 \beta + 5) q^{28} + (3 \beta + 3) q^{29} + ( - 4 \beta + 5) q^{31} + (\beta + 3) q^{32} - 9 q^{34} + (2 \beta + 5) q^{37} + \beta q^{38} + ( - 2 \beta + 3) q^{41} + ( - 4 \beta + 2) q^{43} + (\beta + 1) q^{44} + ( - 5 \beta + 3) q^{46} + 3 q^{47} + 5 \beta q^{49} + (5 \beta + 5) q^{52} - \beta q^{53} + ( - 3 \beta - 6) q^{56} + ( - 6 \beta - 9) q^{58} + ( - 4 \beta + 9) q^{59} + (3 \beta - 4) q^{61} + ( - \beta + 12) q^{62} + ( - 6 \beta + 1) q^{64} - 4 q^{67} + (3 \beta + 6) q^{68} - 2 \beta q^{71} + (3 \beta - 4) q^{73} + ( - 7 \beta - 6) q^{74} + ( - \beta - 1) q^{76} + (\beta + 2) q^{77} + (3 \beta - 7) q^{79} + ( - \beta + 6) q^{82} + ( - 5 \beta - 3) q^{83} + (2 \beta + 12) q^{86} - 3 q^{88} + ( - \beta - 3) q^{89} + (5 \beta + 10) q^{91} + (4 \beta + 3) q^{92} - 3 \beta q^{94} + ( - \beta + 14) q^{97} + ( - 5 \beta - 15) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} + 2 q^{11} + 10 q^{13} - 9 q^{14} - 3 q^{16} - 3 q^{17} - 2 q^{19} - q^{22} + 11 q^{23} - 5 q^{26} + 14 q^{28} + 9 q^{29} + 6 q^{31} + 7 q^{32} - 18 q^{34} + 12 q^{37} + q^{38} + 4 q^{41} + 3 q^{44} + q^{46} + 6 q^{47} + 5 q^{49} + 15 q^{52} - q^{53} - 15 q^{56} - 24 q^{58} + 14 q^{59} - 5 q^{61} + 23 q^{62} - 4 q^{64} - 8 q^{67} + 15 q^{68} - 2 q^{71} - 5 q^{73} - 19 q^{74} - 3 q^{76} + 5 q^{77} - 11 q^{79} + 11 q^{82} - 11 q^{83} + 26 q^{86} - 6 q^{88} - 7 q^{89} + 25 q^{91} + 10 q^{92} - 3 q^{94} + 27 q^{97} - 35 q^{98}+O(q^{100}) \) 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 4.30278 −3.00000 0 0
1.2 1.30278 0 −0.302776 0 0 0.697224 −3.00000 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.o 2
3.b odd 2 1 275.2.a.f yes 2
5.b even 2 1 2475.2.a.t 2
5.c odd 4 2 2475.2.c.k 4
12.b even 2 1 4400.2.a.bh 2
15.d odd 2 1 275.2.a.e 2
15.e even 4 2 275.2.b.c 4
33.d even 2 1 3025.2.a.h 2
60.h even 2 1 4400.2.a.bs 2
60.l odd 4 2 4400.2.b.y 4
165.d even 2 1 3025.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 15.d odd 2 1
275.2.a.f yes 2 3.b odd 2 1
275.2.b.c 4 15.e even 4 2
2475.2.a.o 2 1.a even 1 1 trivial
2475.2.a.t 2 5.b even 2 1
2475.2.c.k 4 5.c odd 4 2
3025.2.a.h 2 33.d even 2 1
3025.2.a.n 2 165.d even 2 1
4400.2.a.bh 2 12.b even 2 1
4400.2.a.bs 2 60.h even 2 1
4400.2.b.y 4 60.l odd 4 2

Hecke kernels

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

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 5T_{7} + 3 \) Copy content Toggle raw display
\( T_{29}^{2} - 9T_{29} - 9 \) 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
$7$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 11T + 27 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 43 \) 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
$43$ \( T^{2} - 52 \) Copy content Toggle raw display
$47$ \( (T - 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T - 3 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 11T - 51 \) Copy content Toggle raw display
$89$ \( T^{2} + 7T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} - 27T + 179 \) Copy content Toggle raw display
show more
show less