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Properties

Label 2340.2.h.f
Level $2340$
Weight $2$
Character orbit 2340.h
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(469,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2340.469");
 
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.h (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: 12.0.2593100598870016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{5} - \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{5} - \beta_{4} q^{7} + (\beta_{8} + \beta_{6} + \beta_{3}) q^{11} + \beta_1 q^{13} + \beta_{2} q^{17} + ( - \beta_{11} - \beta_{7}) q^{19} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_{2}) q^{23}+ \cdots + (3 \beta_{10} + 2 \beta_{4} - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{19} + 4 q^{25} - 16 q^{49} - 12 q^{55} + 4 q^{61} + 20 q^{79} - 16 q^{85} + 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{10} - 4\nu^{8} - \nu^{6} + 8\nu^{4} - 12\nu^{2} - 32 ) / 80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 2\nu^{7} + 5\nu^{5} - 6\nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{11} + 8\nu^{9} - 3\nu^{7} - 36\nu^{5} - 36\nu^{3} - 96\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} - 6\nu^{8} + 11\nu^{6} - 18\nu^{4} + 62\nu^{2} - 88 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{11} + 24\nu^{9} - 9\nu^{7} + 52\nu^{5} - 108\nu^{3} + 192\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{11} - 7\nu^{9} + 12\nu^{7} - 11\nu^{5} + 14\nu^{3} - 56\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} - 2\nu^{8} + \nu^{6} - 6\nu^{4} + 20\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{11} - 8\nu^{9} + 13\nu^{7} - 24\nu^{5} + 86\nu^{3} - 124\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{11} + 12\nu^{9} - 27\nu^{7} + 56\nu^{5} - 104\nu^{3} + 56\nu ) / 80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{10} + 12\nu^{8} - 27\nu^{6} + 16\nu^{4} - 104\nu^{2} + 96 ) / 40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{10} + 3\nu^{8} - 5\nu^{6} + 11\nu^{4} - 20\nu^{2} + 32 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} - \beta_{6} - 2\beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{7} + 2\beta_{4} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{8} - 3\beta_{6} + \beta_{5} - 3\beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} - 3\beta_{10} + \beta_{4} + 6\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{9} + 3\beta_{8} + 3\beta_{6} + 4\beta_{5} - 6\beta_{3} + 3\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{11} - 4\beta_{10} - 9\beta_{7} + 2\beta_{4} - 6\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -4\beta_{9} + 2\beta_{8} + 11\beta_{6} + 19\beta_{5} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2\beta_{11} - \beta_{10} - 4\beta_{7} + 3\beta_{4} - 54\beta _1 - 30 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -15\beta_{9} + 9\beta_{8} - 3\beta_{6} + 24\beta_{5} + 26\beta_{3} + 21\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -3\beta_{11} - 16\beta_{10} + 13\beta_{7} - 30\beta_{4} - 26\beta _1 - 54 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -40\beta_{9} - 6\beta_{8} + 13\beta_{6} - 15\beta_{5} + 29\beta_{3} + 63\beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
1.25694 − 0.648161i
1.25694 + 0.648161i
−1.37820 + 0.317122i
−1.37820 − 0.317122i
0.721581 + 1.21627i
0.721581 − 1.21627i
−0.721581 + 1.21627i
−0.721581 − 1.21627i
1.37820 + 0.317122i
1.37820 − 0.317122i
−1.25694 − 0.648161i
−1.25694 + 0.648161i
0 0 0 −2.15160 − 0.608775i 0 4.25879i 0 0 0
469.2 0 0 0 −2.15160 + 0.608775i 0 − 4.25879i 0 0 0
469.3 0 0 0 −1.45804 − 1.69532i 0 0.748228i 0 0 0
469.4 0 0 0 −1.45804 + 1.69532i 0 − 0.748228i 0 0 0
469.5 0 0 0 −1.11567 − 1.93785i 0 − 2.51056i 0 0 0
469.6 0 0 0 −1.11567 + 1.93785i 0 2.51056i 0 0 0
469.7 0 0 0 1.11567 − 1.93785i 0 2.51056i 0 0 0
469.8 0 0 0 1.11567 + 1.93785i 0 − 2.51056i 0 0 0
469.9 0 0 0 1.45804 − 1.69532i 0 − 0.748228i 0 0 0
469.10 0 0 0 1.45804 + 1.69532i 0 0.748228i 0 0 0
469.11 0 0 0 2.15160 − 0.608775i 0 − 4.25879i 0 0 0
469.12 0 0 0 2.15160 + 0.608775i 0 4.25879i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.h.f ✓ 12
3.b odd 2 1 inner 2340.2.h.f ✓ 12
5.b even 2 1 inner 2340.2.h.f ✓ 12
15.d odd 2 1 inner 2340.2.h.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.h.f ✓ 12 1.a even 1 1 trivial
2340.2.h.f ✓ 12 3.b odd 2 1 inner
2340.2.h.f ✓ 12 5.b even 2 1 inner
2340.2.h.f ✓ 12 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 25T_{7}^{4} + 128T_{7}^{2} + 64 \) 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
$5$ \( T^{12} - 2 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 25 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 33 T^{4} + 144 T^{2} - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$17$ \( (T^{6} + 21 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 2 T^{2} - 32 T - 80)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + 77 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 80 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 40 T + 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 65 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 65 T^{4} + 144 T^{2} - 4)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$47$ \( (T^{6} + 144 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 77 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 236 T^{4} + \cdots - 102400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - T^{2} - 24 T - 32)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 80 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 257 T^{4} + \cdots - 188356)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{6} \) Copy content Toggle raw display
$79$ \( (T^{3} - 5 T^{2} + \cdots + 284)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 272 T^{4} + \cdots + 355216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 137 T^{4} + \cdots - 45796)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 497 T^{4} + \cdots + 163216)^{2} \) Copy content Toggle raw display
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