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Properties

Label 2394.2.b.j
Level $2394$
Weight $2$
Character orbit 2394.b
Analytic conductor $19.116$
Analytic rank $0$
Dimension $8$
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] = [2394,2,Mod(1709,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.1709");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.b (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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 30x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_{3} q^{5} - q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta_{3} q^{5} - q^{7} + q^{8} + \beta_{3} q^{10} - \beta_{5} q^{11} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{13} - q^{14} + q^{16} - \beta_{5} q^{17} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{19} + \beta_{3} q^{20} - \beta_{5} q^{22} + (\beta_{5} - \beta_{2}) q^{23} + (\beta_{7} + \beta_{4} - 3) q^{25} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{26} - q^{28} + (\beta_{7} + \beta_{4} - 2) q^{29} + ( - \beta_{5} - \beta_{3}) q^{31} + q^{32} - \beta_{5} q^{34} - \beta_{3} q^{35} + (2 \beta_{3} - \beta_{2}) q^{37} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{38} + \beta_{3} q^{40} + (2 \beta_{6} - 2 \beta_{4} - \beta_1 + 5) q^{41} + ( - \beta_{6} + \beta_{4} - \beta_1 - 5) q^{43} - \beta_{5} q^{44} + (\beta_{5} - \beta_{2}) q^{46} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{47} + q^{49} + (\beta_{7} + \beta_{4} - 3) q^{50} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{52} + ( - \beta_{7} - \beta_{4} - 2 \beta_1 + 6) q^{53} + ( - \beta_{7} - \beta_{6} - 3 \beta_1 - 1) q^{55} - q^{56} + (\beta_{7} + \beta_{4} - 2) q^{58} + ( - 2 \beta_{7} - 2 \beta_{4} + \cdots + 1) q^{61}+ \cdots + q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8} - 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} - 16 q^{29} + 8 q^{32} + 12 q^{38} + 24 q^{41} - 32 q^{43} + 8 q^{49} - 24 q^{50} + 48 q^{53} - 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} + 16 q^{65} - 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} - 32 q^{86} - 8 q^{89} + 8 q^{95} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 30x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{4} + 10\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\nu^{5} - 18\nu^{3} - 21\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} + 8\nu^{5} + 19\nu^{3} + 14\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{7} + \nu^{6} + 10\nu^{5} + 5\nu^{4} + 29\nu^{3} + 3\nu^{2} + 20\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} - 11\nu^{5} - 35\nu^{3} - 25\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - \nu^{6} + 10\nu^{5} - 7\nu^{4} + 29\nu^{3} - 11\nu^{2} + 20\nu - 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} - \nu^{6} - 10\nu^{5} - 7\nu^{4} - 29\nu^{3} - 11\nu^{2} - 20\nu - 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} + 6\beta_{5} - 2\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{4} + \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{7} - 15\beta_{6} - 24\beta_{5} + 6\beta_{3} + 4\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{7} - 5\beta_{4} - 4\beta _1 + 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -69\beta_{7} + 69\beta_{6} + 102\beta_{5} - 36\beta_{3} - 14\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23\beta_{7} - \beta_{6} + 24\beta_{4} + 17\beta _1 - 87 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 309\beta_{7} - 309\beta_{6} - 444\beta_{5} + 186\beta_{3} + 64\beta_{2} ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1709.1
2.15177i
− 2.03530i
− 0.209899i
− 1.08784i
1.08784i
0.209899i
2.03530i
− 2.15177i
1.00000 0 1.00000 − 3.71953i 0 −1.00000 1.00000 0 − 3.71953i
1709.2 1.00000 0 1.00000 − 3.02986i 0 −1.00000 1.00000 0 − 3.02986i
1709.3 1.00000 0 1.00000 − 2.76612i 0 −1.00000 1.00000 0 − 2.76612i
1709.4 1.00000 0 1.00000 − 1.15484i 0 −1.00000 1.00000 0 − 1.15484i
1709.5 1.00000 0 1.00000 1.15484i 0 −1.00000 1.00000 0 1.15484i
1709.6 1.00000 0 1.00000 2.76612i 0 −1.00000 1.00000 0 2.76612i
1709.7 1.00000 0 1.00000 3.02986i 0 −1.00000 1.00000 0 3.02986i
1709.8 1.00000 0 1.00000 3.71953i 0 −1.00000 1.00000 0 3.71953i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1709.8
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 2394.2.b.j yes 8
3.b odd 2 1 2394.2.b.i ✓ 8
19.b odd 2 1 2394.2.b.i ✓ 8
57.d even 2 1 inner 2394.2.b.j yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.i ✓ 8 3.b odd 2 1
2394.2.b.i ✓ 8 19.b odd 2 1
2394.2.b.j yes 8 1.a even 1 1 trivial
2394.2.b.j yes 8 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5}^{8} + 32T_{5}^{6} + 344T_{5}^{4} + 1376T_{5}^{2} + 1296 \) Copy content Toggle raw display
\( T_{29}^{4} + 8T_{29}^{3} - 16T_{29}^{2} - 160T_{29} - 192 \) Copy content Toggle raw display
\( T_{53}^{4} - 24T_{53}^{3} + 152T_{53}^{2} + 128T_{53} - 2544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 32 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 56 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{8} + 72 T^{6} + \cdots + 14400 \) Copy content Toggle raw display
$17$ \( T^{8} + 56 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 80 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} + \cdots - 192)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 88 T^{6} + \cdots + 32400 \) Copy content Toggle raw display
$37$ \( T^{8} + 152 T^{6} + \cdots + 138384 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} + \cdots - 864)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 288 T^{6} + \cdots + 3686400 \) Copy content Toggle raw display
$53$ \( (T^{4} - 24 T^{3} + \cdots - 2544)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 2144)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 304 T^{6} + \cdots + 10419984 \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} + \cdots + 768)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots + 2480)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 360 T^{6} + \cdots + 18455616 \) Copy content Toggle raw display
$83$ \( T^{8} + 424 T^{6} + \cdots + 68757264 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots + 9312)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 288 T^{6} + \cdots + 2322576 \) Copy content Toggle raw display
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