Properties

Label 2310.2.a.z
Level $2310$
Weight $2$
Character orbit 2310.a
Self dual yes
Analytic conductor $18.445$
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] = [2310,2,Mod(1,2310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.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: \(18.4454428669\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + \beta q^{13} + q^{14} + q^{15} + q^{16} + \beta q^{17} - q^{18} + q^{20} - q^{21} - q^{22} + (\beta + 2) q^{23} - q^{24} + q^{25} - \beta q^{26} + q^{27} - q^{28} + ( - 2 \beta + 2) q^{29} - q^{30} + ( - 2 \beta + 4) q^{31} - q^{32} + q^{33} - \beta q^{34} - q^{35} + q^{36} - \beta q^{37} + \beta q^{39} - q^{40} + 2 q^{41} + q^{42} + q^{44} + q^{45} + ( - \beta - 2) q^{46} - 2 \beta q^{47} + q^{48} + q^{49} - q^{50} + \beta q^{51} + \beta q^{52} + 6 q^{53} - q^{54} + q^{55} + q^{56} + (2 \beta - 2) q^{58} + 8 q^{59} + q^{60} + (2 \beta + 2) q^{61} + (2 \beta - 4) q^{62} - q^{63} + q^{64} + \beta q^{65} - q^{66} + (\beta - 2) q^{67} + \beta q^{68} + (\beta + 2) q^{69} + q^{70} + (2 \beta + 8) q^{71} - q^{72} + ( - 4 \beta + 2) q^{73} + \beta q^{74} + q^{75} - q^{77} - \beta q^{78} - 4 q^{79} + q^{80} + q^{81} - 2 q^{82} - 4 \beta q^{83} - q^{84} + \beta q^{85} + ( - 2 \beta + 2) q^{87} - q^{88} + (\beta + 12) q^{89} - q^{90} - \beta q^{91} + (\beta + 2) q^{92} + ( - 2 \beta + 4) q^{93} + 2 \beta q^{94} - q^{96} - 10 q^{97} - q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 2 q^{21} - 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} + 8 q^{31} - 2 q^{32} + 2 q^{33} - 2 q^{35} + 2 q^{36} - 2 q^{40} + 4 q^{41} + 2 q^{42} + 2 q^{44} + 2 q^{45} - 4 q^{46} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 12 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{56} - 4 q^{58} + 16 q^{59} + 2 q^{60} + 4 q^{61} - 8 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{66} - 4 q^{67} + 4 q^{69} + 2 q^{70} + 16 q^{71} - 2 q^{72} + 4 q^{73} + 2 q^{75} - 2 q^{77} - 8 q^{79} + 2 q^{80} + 2 q^{81} - 4 q^{82} - 2 q^{84} + 4 q^{87} - 2 q^{88} + 24 q^{89} - 2 q^{90} + 4 q^{92} + 8 q^{93} - 2 q^{96} - 20 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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
−1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 2310.2.a.z 2
3.b odd 2 1 6930.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.z 2 1.a even 1 1 trivial
6930.2.a.bw 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2310))\):

\( T_{13}^{2} - 12 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 8 \) Copy content Toggle raw display
\( T_{29}^{2} - 4T_{29} - 44 \) Copy content Toggle raw display
\( T_{31}^{2} - 8T_{31} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 12 \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$37$ \( T^{2} - 12 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 48 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 188 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 24T + 132 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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