Properties

Label 2475.2.c.i
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(199,2475)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} - \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} - \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8} - q^{11} + ( - 3 \beta_{2} - \beta_1) q^{13} + (\beta_{3} - 1) q^{14} + ( - 3 \beta_{3} + 3) q^{16} + ( - 4 \beta_{2} - \beta_1) q^{17} - 3 q^{19} - \beta_1 q^{22} + (4 \beta_{2} - \beta_1) q^{23} + (2 \beta_{3} + 2) q^{26} + (2 \beta_{2} - \beta_1) q^{28} + (2 \beta_{3} - 2) q^{29} - 3 \beta_{3} q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} + (3 \beta_{3} + 1) q^{34} + ( - 2 \beta_{2} - 5 \beta_1) q^{37} - 3 \beta_1 q^{38} + (3 \beta_{3} - 3) q^{41} + ( - 3 \beta_{2} - 3 \beta_1) q^{43} + ( - \beta_{3} + 3) q^{44} + ( - 5 \beta_{3} + 9) q^{46} + ( - 8 \beta_{2} - \beta_1) q^{47} + 6 q^{49} + 2 \beta_{2} q^{52} + (4 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{3} + 5) q^{56} + (8 \beta_{2} - 2 \beta_1) q^{58} + (\beta_{3} - 9) q^{59} + (\beta_{3} - 6) q^{61} - 12 \beta_{2} q^{62} + ( - \beta_{3} - 3) q^{64} + ( - \beta_{2} - 3 \beta_1) q^{67} + (4 \beta_{2} - \beta_1) q^{68} + ( - 5 \beta_{3} + 1) q^{71} + (6 \beta_{2} - 4 \beta_1) q^{73} + ( - 3 \beta_{3} + 23) q^{74} + ( - 3 \beta_{3} + 9) q^{76} + \beta_{2} q^{77} + ( - 3 \beta_{3} + 11) q^{79} + (12 \beta_{2} - 3 \beta_1) q^{82} + 12 q^{86} + ( - 4 \beta_{2} + \beta_1) q^{88} + ( - 2 \beta_{3} + 14) q^{89} + ( - \beta_{3} - 2) q^{91} + ( - 12 \beta_{2} + 7 \beta_1) q^{92} + (7 \beta_{3} - 3) q^{94} + (3 \beta_{2} + 2 \beta_1) q^{97} + 6 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} - 4 q^{11} - 2 q^{14} + 6 q^{16} - 12 q^{19} + 12 q^{26} - 4 q^{29} - 6 q^{31} + 10 q^{34} - 6 q^{41} + 10 q^{44} + 26 q^{46} + 24 q^{49} + 18 q^{56} - 34 q^{59} - 22 q^{61} - 14 q^{64} - 6 q^{71} + 86 q^{74} + 30 q^{76} + 38 q^{79} + 48 q^{86} + 52 q^{89} - 10 q^{91} + 2 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 0
199.2 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.3 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.4 2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{29}^{2} + 2T_{29} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 33T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T + 3)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 49T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T - 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 213 T^{2} + 11236 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T - 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 81T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( T^{4} + 121T^{2} + 2704 \) Copy content Toggle raw display
$53$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$59$ \( (T^{2} + 17 T + 68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 26)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 77T^{2} + 1444 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3 T - 104)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 264T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 19 T + 52)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 26 T + 152)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 42T^{2} + 169 \) Copy content Toggle raw display
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