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Properties

Label 456.2.f.a
Level $456$
Weight $2$
Character orbit 456.f
Analytic conductor $3.641$
Analytic rank $0$
Dimension $10$
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] = [456,2,Mod(113,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.f (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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.20322144469993472.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - x^{8} - 2x^{7} - 2x^{6} + 22x^{5} - 6x^{4} - 18x^{3} - 27x^{2} - 81x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{6} q^{5} - \beta_{4} q^{7} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + \beta_{6} q^{5} - \beta_{4} q^{7} - \beta_{3} q^{9} - \beta_{9} q^{11} + ( - \beta_{9} - \beta_{6} + \cdots - \beta_1) q^{13}+ \cdots + ( - 2 \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{3} - 2 q^{7} + 3 q^{9} + 10 q^{15} + 2 q^{19} + 5 q^{21} - 14 q^{25} - 10 q^{27} - 6 q^{29} - 10 q^{33} + 7 q^{39} - 4 q^{41} + 20 q^{43} - 2 q^{45} + 16 q^{49} + q^{51} - 26 q^{53} - 12 q^{55} + 9 q^{57} + 2 q^{59} - 4 q^{61} - 17 q^{63} + 32 q^{65} - 27 q^{69} + 8 q^{71} - 26 q^{73} + 39 q^{75} + 23 q^{81} - 8 q^{85} + 13 q^{87} + 4 q^{89} - 18 q^{93} - 8 q^{95} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - x^{8} - 2x^{7} - 2x^{6} + 22x^{5} - 6x^{4} - 18x^{3} - 27x^{2} - 81x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 2\nu^{8} - 4\nu^{7} - 5\nu^{6} - 8\nu^{5} + 16\nu^{4} + 60\nu^{3} - 36\nu^{2} - 81\nu - 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{9} - 4\nu^{8} - \nu^{7} + 19\nu^{6} + 25\nu^{5} - 14\nu^{4} - 21\nu^{3} - 45\nu^{2} + 135\nu + 324 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - \nu^{8} - \nu^{7} - 2\nu^{6} - 2\nu^{5} + 22\nu^{4} - 6\nu^{3} - 18\nu^{2} - 27\nu - 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{9} - 4\nu^{8} + 5\nu^{7} + \nu^{6} - 17\nu^{5} - 47\nu^{4} - 15\nu^{3} + 81\nu^{2} + 81 ) / 324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{9} - 5\nu^{8} - 8\nu^{7} + 8\nu^{6} + 11\nu^{5} - 13\nu^{4} - 60\nu^{3} - 54\nu^{2} + 135\nu + 162 ) / 162 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{9} + 2\nu^{8} - 7\nu^{7} - 14\nu^{6} + 40\nu^{5} + \nu^{4} + 3\nu^{3} - 108\nu^{2} - 162\nu + 405 ) / 162 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{9} + 2\nu^{8} - 7\nu^{7} + 13\nu^{6} + 13\nu^{5} - 26\nu^{4} - 51\nu^{3} - 81\nu^{2} + 27\nu + 324 ) / 162 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 2\beta_{7} + 4\beta_{5} + \beta_{4} + \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} + 2\beta_{8} - 4\beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{9} - 4\beta_{8} - 2\beta_{7} - 4\beta_{6} + 5\beta_{4} + \beta_{3} + \beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{9} - 2\beta_{8} - 6\beta_{7} - 8\beta_{6} - 5\beta_{5} + 2\beta_{4} - 7\beta_{3} - 9\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{9} - 2 \beta_{8} - 12 \beta_{7} - 20 \beta_{6} - 6 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 19 \beta_{9} - 8 \beta_{8} + 22 \beta_{7} - 44 \beta_{6} - 28 \beta_{5} - 9 \beta_{4} - 4 \beta_{3} + \cdots + 29 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(343\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
1.69830 − 0.340259i
1.69830 + 0.340259i
1.39399 − 1.02801i
1.39399 + 1.02801i
−0.171092 − 1.72358i
−0.171092 + 1.72358i
−0.729858 − 1.57077i
−0.729858 + 1.57077i
−1.69134 − 0.373340i
−1.69134 + 0.373340i
0 −1.69830 − 0.340259i 0 3.78858i 0 −4.37856 0 2.76845 + 1.15572i 0
113.2 0 −1.69830 + 0.340259i 0 − 3.78858i 0 −4.37856 0 2.76845 − 1.15572i 0
113.3 0 −1.39399 − 1.02801i 0 − 1.12291i 0 3.21832 0 0.886389 + 2.86606i 0
113.4 0 −1.39399 + 1.02801i 0 1.12291i 0 3.21832 0 0.886389 − 2.86606i 0
113.5 0 0.171092 − 1.72358i 0 3.81594i 0 2.25057 0 −2.94146 − 0.589781i 0
113.6 0 0.171092 + 1.72358i 0 − 3.81594i 0 2.25057 0 −2.94146 + 0.589781i 0
113.7 0 0.729858 − 1.57077i 0 − 1.22502i 0 −2.80880 0 −1.93462 − 2.29287i 0
113.8 0 0.729858 + 1.57077i 0 1.22502i 0 −2.80880 0 −1.93462 + 2.29287i 0
113.9 0 1.69134 − 0.373340i 0 0.568907i 0 0.718465 0 2.72123 − 1.26289i 0
113.10 0 1.69134 + 0.373340i 0 − 0.568907i 0 0.718465 0 2.72123 + 1.26289i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.f.a ✓ 10
3.b odd 2 1 456.2.f.b yes 10
4.b odd 2 1 912.2.f.i 10
12.b even 2 1 912.2.f.h 10
19.b odd 2 1 456.2.f.b yes 10
57.d even 2 1 inner 456.2.f.a ✓ 10
76.d even 2 1 912.2.f.h 10
228.b odd 2 1 912.2.f.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.f.a ✓ 10 1.a even 1 1 trivial
456.2.f.a ✓ 10 57.d even 2 1 inner
456.2.f.b yes 10 3.b odd 2 1
456.2.f.b yes 10 19.b odd 2 1
912.2.f.h 10 12.b even 2 1
912.2.f.h 10 76.d even 2 1
912.2.f.i 10 4.b odd 2 1
912.2.f.i 10 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{29}^{5} + 3T_{29}^{4} - 52T_{29}^{3} - 140T_{29}^{2} + 608T_{29} + 1216 \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + 32 T^{8} + \cdots + 128 \) Copy content Toggle raw display
$7$ \( (T^{5} + T^{4} - 21 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 50 T^{8} + \cdots + 41472 \) Copy content Toggle raw display
$13$ \( T^{10} + 57 T^{8} + \cdots + 128 \) Copy content Toggle raw display
$17$ \( T^{10} + 101 T^{8} + \cdots + 524288 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} + 129 T^{8} + \cdots + 46208 \) Copy content Toggle raw display
$29$ \( (T^{5} + 3 T^{4} + \cdots + 1216)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 134 T^{8} + \cdots + 107648 \) Copy content Toggle raw display
$37$ \( T^{10} + 174 T^{8} + \cdots + 20891648 \) Copy content Toggle raw display
$41$ \( (T^{5} + 2 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 10 T^{4} + \cdots + 432)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 204 T^{8} + \cdots + 30752 \) Copy content Toggle raw display
$53$ \( (T^{5} + 13 T^{4} + \cdots + 1432)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - T^{4} + \cdots + 18336)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 2 T^{4} + \cdots + 3488)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 757071872 \) Copy content Toggle raw display
$71$ \( (T^{5} - 4 T^{4} + \cdots - 7936)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 13 T^{4} + \cdots - 13752)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 486 T^{8} + \cdots + 11829248 \) Copy content Toggle raw display
$83$ \( T^{10} + 240 T^{8} + \cdots + 2654208 \) Copy content Toggle raw display
$89$ \( (T^{5} - 2 T^{4} + \cdots - 432)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 169869312 \) Copy content Toggle raw display
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