Properties

Label 2115.2.a.l
Level $2115$
Weight $2$
Character orbit 2115.a
Self dual yes
Analytic conductor $16.888$
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] = [2115,2,Mod(1,2115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2115, 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("2115.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2115 = 3^{2} \cdot 5 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2115.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: \(16.8883600275\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 705)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} + q^{5} - q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{4} + q^{5} - q^{7} - \beta q^{8} + \beta q^{10} + 4 q^{11} + (\beta + 4) q^{13} - \beta q^{14} - 5 q^{16} + ( - 3 \beta + 2) q^{17} + \beta q^{19} + q^{20} + 4 \beta q^{22} + (2 \beta + 1) q^{23} + q^{25} + (4 \beta + 3) q^{26} - q^{28} + ( - 2 \beta + 3) q^{29} - 2 \beta q^{31} - 3 \beta q^{32} + (2 \beta - 9) q^{34} - q^{35} + (2 \beta + 8) q^{37} + 3 q^{38} - \beta q^{40} + (2 \beta + 3) q^{41} + (2 \beta - 8) q^{43} + 4 q^{44} + (\beta + 6) q^{46} + q^{47} - 6 q^{49} + \beta q^{50} + (\beta + 4) q^{52} + (7 \beta - 2) q^{53} + 4 q^{55} + \beta q^{56} + (3 \beta - 6) q^{58} + (3 \beta + 10) q^{59} + ( - 4 \beta + 1) q^{61} - 6 q^{62} + q^{64} + (\beta + 4) q^{65} + ( - 4 \beta - 2) q^{67} + ( - 3 \beta + 2) q^{68} - \beta q^{70} + ( - 3 \beta + 2) q^{71} + 4 q^{73} + (8 \beta + 6) q^{74} + \beta q^{76} - 4 q^{77} + (2 \beta + 2) q^{79} - 5 q^{80} + (3 \beta + 6) q^{82} + (4 \beta - 4) q^{83} + ( - 3 \beta + 2) q^{85} + ( - 8 \beta + 6) q^{86} - 4 \beta q^{88} + ( - 2 \beta + 10) q^{89} + ( - \beta - 4) q^{91} + (2 \beta + 1) q^{92} + \beta q^{94} + \beta q^{95} + ( - 2 \beta - 2) q^{97} - 6 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{5} - 2 q^{7} + 8 q^{11} + 8 q^{13} - 10 q^{16} + 4 q^{17} + 2 q^{20} + 2 q^{23} + 2 q^{25} + 6 q^{26} - 2 q^{28} + 6 q^{29} - 18 q^{34} - 2 q^{35} + 16 q^{37} + 6 q^{38} + 6 q^{41} - 16 q^{43} + 8 q^{44} + 12 q^{46} + 2 q^{47} - 12 q^{49} + 8 q^{52} - 4 q^{53} + 8 q^{55} - 12 q^{58} + 20 q^{59} + 2 q^{61} - 12 q^{62} + 2 q^{64} + 8 q^{65} - 4 q^{67} + 4 q^{68} + 4 q^{71} + 8 q^{73} + 12 q^{74} - 8 q^{77} + 4 q^{79} - 10 q^{80} + 12 q^{82} - 8 q^{83} + 4 q^{85} + 12 q^{86} + 20 q^{89} - 8 q^{91} + 2 q^{92} - 4 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 0 1.00000 1.00000 0 −1.00000 1.73205 0 −1.73205
1.2 1.73205 0 1.00000 1.00000 0 −1.00000 −1.73205 0 1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(47\) \(-1\)

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 2115.2.a.l 2
3.b odd 2 1 705.2.a.i 2
15.d odd 2 1 3525.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.i 2 3.b odd 2 1
2115.2.a.l 2 1.a even 1 1 trivial
3525.2.a.p 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2115))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 23 \) Copy content Toggle raw display
$19$ \( T^{2} - 3 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 3 \) Copy content Toggle raw display
$31$ \( T^{2} - 12 \) Copy content Toggle raw display
$37$ \( T^{2} - 16T + 52 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 3 \) Copy content Toggle raw display
$43$ \( T^{2} + 16T + 52 \) Copy content Toggle raw display
$47$ \( (T - 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 143 \) Copy content Toggle raw display
$59$ \( T^{2} - 20T + 73 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 47 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 23 \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 20T + 88 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
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