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Properties

Label 2340.2.bi.d
Level $2340$
Weight $2$
Character orbit 2340.bi
Analytic conductor $18.685$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2340,2,Mod(161,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2340.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.bi (of order \(4\), degree \(2\), 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: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.58498535041007616.52
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} - \beta_{11} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} - \beta_{11} q^{7} + ( - \beta_{10} - 2 \beta_1) q^{11} + ( - \beta_{4} - \beta_{2} + 1) q^{13} + (\beta_{8} + \beta_{3} + \beta_1) q^{17} + (2 \beta_{5} + 2) q^{19} + (\beta_{10} - \beta_{8} + \cdots + 2 \beta_1) q^{23}+ \cdots + (\beta_{9} + 3 \beta_{5} + \beta_{4} + \cdots + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{13} + 24 q^{19} - 12 q^{37} - 24 q^{55} - 12 q^{73} + 24 q^{79} + 12 q^{85} + 72 q^{91} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 3\nu^{6} + 18\nu^{3} - 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 6\nu^{10} - 3\nu^{8} + 45\nu^{7} + 45\nu^{5} - 189\nu^{4} - 162\nu^{2} + 729\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{9} + 15\nu^{6} - 63\nu^{3} + 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + 6\nu^{10} + 12\nu^{8} - 45\nu^{7} - 72\nu^{5} + 270\nu^{4} + 324\nu^{2} - 972\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{9} - 15\nu^{6} + 90\nu^{3} - 324 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} - 6\nu^{10} + 3\nu^{8} + 45\nu^{7} - 45\nu^{5} - 189\nu^{4} + 162\nu^{2} + 729\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} - 6\nu^{10} + 12\nu^{8} + 45\nu^{7} - 72\nu^{5} - 270\nu^{4} + 324\nu^{2} + 972\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + \nu^{10} + 6\nu^{8} - 3\nu^{7} - 27\nu^{5} + 18\nu^{4} + 135\nu^{2} - 81\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} - \nu^{10} + 6\nu^{8} + 3\nu^{7} - 27\nu^{5} - 18\nu^{4} + 135\nu^{2} + 81\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4\nu^{11} - 30\nu^{8} + 153\nu^{5} - 486\nu^{2} + 243\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -4\nu^{11} + 30\nu^{8} - 153\nu^{5} + 486\nu^{2} + 243\nu ) / 243 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} + 3\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{11} + 3\beta_{10} - 3\beta_{7} + 3\beta_{6} + 3\beta_{4} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{11} + 3\beta_{10} + 6\beta_{9} + 6\beta_{8} - 6\beta_{6} + 6\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{5} + 18\beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -18\beta_{9} + 18\beta_{8} - 9\beta_{7} + 18\beta_{6} + 9\beta_{4} + 18\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9\beta_{11} - 9\beta_{10} + 9\beta_{7} - 45\beta_{6} + 9\beta_{4} + 45\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -27\beta_{5} + 135\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 27\beta_{11} + 27\beta_{10} - 135\beta_{9} + 135\beta_{8} + 27\beta_{7} - 27\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -108\beta_{9} - 108\beta_{8} + 189\beta_{7} - 108\beta_{6} + 189\beta_{4} + 108\beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(1\) \(\beta_{5}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.0980500 + 1.72927i
1.54662 − 0.779723i
−1.44857 − 0.949550i
1.72927 − 0.0980500i
−0.779723 + 1.54662i
−0.949550 − 1.44857i
−0.0980500 − 1.72927i
1.54662 + 0.779723i
−1.44857 + 0.949550i
1.72927 + 0.0980500i
−0.779723 − 1.54662i
−0.949550 + 1.44857i
0 0 0 −0.707107 − 0.707107i 0 −1.63122 − 1.63122i 0 0 0
161.2 0 0 0 −0.707107 − 0.707107i 0 −0.766897 − 0.766897i 0 0 0
161.3 0 0 0 −0.707107 − 0.707107i 0 2.39812 + 2.39812i 0 0 0
161.4 0 0 0 0.707107 + 0.707107i 0 −1.63122 − 1.63122i 0 0 0
161.5 0 0 0 0.707107 + 0.707107i 0 −0.766897 − 0.766897i 0 0 0
161.6 0 0 0 0.707107 + 0.707107i 0 2.39812 + 2.39812i 0 0 0
1061.1 0 0 0 −0.707107 + 0.707107i 0 −1.63122 + 1.63122i 0 0 0
1061.2 0 0 0 −0.707107 + 0.707107i 0 −0.766897 + 0.766897i 0 0 0
1061.3 0 0 0 −0.707107 + 0.707107i 0 2.39812 − 2.39812i 0 0 0
1061.4 0 0 0 0.707107 − 0.707107i 0 −1.63122 + 1.63122i 0 0 0
1061.5 0 0 0 0.707107 − 0.707107i 0 −0.766897 + 0.766897i 0 0 0
1061.6 0 0 0 0.707107 − 0.707107i 0 2.39812 − 2.39812i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.bi.d ✓ 12
3.b odd 2 1 inner 2340.2.bi.d ✓ 12
13.d odd 4 1 inner 2340.2.bi.d ✓ 12
39.f even 4 1 inner 2340.2.bi.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.bi.d ✓ 12 1.a even 1 1 trivial
2340.2.bi.d ✓ 12 3.b odd 2 1 inner
2340.2.bi.d ✓ 12 13.d odd 4 1 inner
2340.2.bi.d ✓ 12 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 12T_{7}^{3} + 81T_{7}^{2} + 108T_{7} + 72 \) acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{6} + 12 T^{3} + \cdots + 72)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 498 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 24 T^{4} + \cdots - 338)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 8)^{6} \) Copy content Toggle raw display
$23$ \( (T^{6} - 78 T^{4} + \cdots - 7688)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 42 T^{4} + \cdots + 1568)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 96 T^{3} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots + 242)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 342102016 \) Copy content Toggle raw display
$43$ \( (T^{6} + 216 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 3782742016 \) Copy content Toggle raw display
$53$ \( (T^{6} + 168 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( (T^{3} - 99 T - 354)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 576 T^{3} + \cdots + 165888)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2018854506496 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 6 T^{2} + \cdots + 112)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + 3720 T^{8} + \cdots + 1048576 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 5972816656 \) Copy content Toggle raw display
$97$ \( (T^{6} - 18 T^{5} + \cdots + 18)^{2} \) Copy content Toggle raw display
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