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Properties

Label 1848.2.v.b
Level $1848$
Weight $2$
Character orbit 1848.v
Analytic conductor $14.756$
Analytic rank $0$
Dimension $8$
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] = [1848,2,Mod(881,1848)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1848, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("1848.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1848.v (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: \(14.7563542935\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5}) q^{3} + \beta_{2} q^{5} + ( - \beta_{7} + \beta_{6} - 1) q^{7} + (\beta_{7} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5}) q^{3} + \beta_{2} q^{5} + ( - \beta_{7} + \beta_{6} - 1) q^{7} + (\beta_{7} - \beta_1) q^{9} + \beta_{3} q^{11} + \beta_{6} q^{13} + ( - 2 \beta_{3} + \beta_1 - 1) q^{15} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{17} + (3 \beta_{6} + 2 \beta_{5} + 2 \beta_{4}) q^{19} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots + 1) q^{21}+ \cdots + (\beta_{7} - \beta_{3} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} - 4 q^{9} - 8 q^{15} + 12 q^{21} - 20 q^{25} + 52 q^{37} + 12 q^{39} - 64 q^{43} + 16 q^{49} - 24 q^{51} + 44 q^{57} - 32 q^{63} + 60 q^{67} - 8 q^{85} - 20 q^{91} + 40 q^{93} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 10\nu^{3} + 24\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{3} + 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 3\nu^{5} + 3\nu^{4} - 2\nu^{3} + 10\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} + 10\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + \nu^{4} - 6\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} - \nu^{4} - 4 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} + 3\beta_{3} - 3\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 5\beta_{6} + 3\beta_{5} + 3\beta_{4} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{5} - 3\beta_{4} + 5\beta_{3} + 3\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{7} - 5\beta_{6} - 3\beta_{5} - 3\beta_{4} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{5} - 5\beta_{4} - 21\beta_{3} - 3\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times\).

\(n\) \(463\) \(617\) \(673\) \(925\) \(1585\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−1.17915 − 0.780776i
−1.17915 + 0.780776i
0.599676 − 1.28078i
0.599676 + 1.28078i
−0.599676 + 1.28078i
−0.599676 − 1.28078i
1.17915 + 0.780776i
1.17915 − 0.780776i
0 −1.51022 − 0.848071i 0 0.662153 0 −2.56155 − 0.662153i 0 1.56155 + 2.56155i 0
881.2 0 −1.51022 + 0.848071i 0 0.662153 0 −2.56155 + 0.662153i 0 1.56155 − 2.56155i 0
881.3 0 −0.468213 − 1.66757i 0 2.13578 0 1.56155 + 2.13578i 0 −2.56155 + 1.56155i 0
881.4 0 −0.468213 + 1.66757i 0 2.13578 0 1.56155 − 2.13578i 0 −2.56155 − 1.56155i 0
881.5 0 0.468213 − 1.66757i 0 −2.13578 0 1.56155 + 2.13578i 0 −2.56155 − 1.56155i 0
881.6 0 0.468213 + 1.66757i 0 −2.13578 0 1.56155 − 2.13578i 0 −2.56155 + 1.56155i 0
881.7 0 1.51022 − 0.848071i 0 −0.662153 0 −2.56155 − 0.662153i 0 1.56155 − 2.56155i 0
881.8 0 1.51022 + 0.848071i 0 −0.662153 0 −2.56155 + 0.662153i 0 1.56155 + 2.56155i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1848.2.v.b ✓ 8
3.b odd 2 1 inner 1848.2.v.b ✓ 8
7.b odd 2 1 inner 1848.2.v.b ✓ 8
21.c even 2 1 inner 1848.2.v.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.2.v.b ✓ 8 1.a even 1 1 trivial
1848.2.v.b ✓ 8 3.b odd 2 1 inner
1848.2.v.b ✓ 8 7.b odd 2 1 inner
1848.2.v.b ✓ 8 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 5T_{5}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1848, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 2 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 37 T^{2} + 338)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 36 T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 13 T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 184 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 13 T + 38)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 112 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 95 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 68)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 173 T^{2} + 338)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{2} + 32)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 15 T + 52)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 79 T^{2} + 1352)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 68)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 250 T^{2} + 5000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 160 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 40 T^{2} + 128)^{2} \) Copy content Toggle raw display
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