Properties

Label 1368.2.f.d
Level $1368$
Weight $2$
Character orbit 1368.f
Analytic conductor $10.924$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,2,Mod(1025,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, 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("1368.1025");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.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: \(10.9235349965\)
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: \( x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{5} + \beta_{4} q^{7} + (\beta_{5} + \beta_{3}) q^{11} + (\beta_{5} - \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} - \beta_{3}) q^{17} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{19} + (\beta_{3} - \beta_{2}) q^{23} + (\beta_{6} - \beta_{4} - \beta_1 - 1) q^{25} + (\beta_{6} - \beta_1 + 4) q^{29} + ( - \beta_{5} - 3 \beta_{3} + \beta_{2}) q^{31} + (\beta_{7} - 2 \beta_{5} + \cdots + \beta_{2}) q^{35}+ \cdots + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{19} - 8 q^{25} + 32 q^{29} - 24 q^{41} - 28 q^{43} + 4 q^{49} + 8 q^{53} + 12 q^{55} + 8 q^{59} - 8 q^{61} - 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} + 16 q^{89} + 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{6} + 17\nu^{4} + 20\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 18\nu^{5} + 37\nu^{3} + 24\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\nu^{7} - 70\nu^{5} - 113\nu^{3} - 32\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 5\nu^{6} + 88\nu^{4} + 150\nu^{2} + 50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} - 158\nu^{5} - 263\nu^{3} - 84\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 7\nu^{6} + 123\nu^{4} + 206\nu^{2} + 65 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\nu^{7} + 264\nu^{5} + 449\nu^{3} + 144\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{6} + 4\beta_{4} + \beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 12\beta_{5} + 9\beta_{3} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{6} - 33\beta_{4} - 10\beta _1 + 55 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35\beta_{7} + 194\beta_{5} - 141\beta_{3} + 93\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -395\beta_{6} + 521\beta_{4} + 161\beta _1 - 858 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -556\beta_{7} - 3060\beta_{5} + 2217\beta_{3} - 1462\beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
1.18847i
0.676777i
0.626815i
3.96694i
3.96694i
0.626815i
0.676777i
1.18847i
0 0 0 3.42865i 0 −0.394100 0 0 0
1025.2 0 0 0 2.60938i 0 −0.723074 0 0 0
1025.3 0 0 0 2.32945i 0 4.34658 0 0 0
1025.4 0 0 0 0.0959660i 0 −3.22941 0 0 0
1025.5 0 0 0 0.0959660i 0 −3.22941 0 0 0
1025.6 0 0 0 2.32945i 0 4.34658 0 0 0
1025.7 0 0 0 2.60938i 0 −0.723074 0 0 0
1025.8 0 0 0 3.42865i 0 −0.394100 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.8
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 1368.2.f.d yes 8
3.b odd 2 1 1368.2.f.c 8
4.b odd 2 1 2736.2.f.h 8
12.b even 2 1 2736.2.f.g 8
19.b odd 2 1 1368.2.f.c 8
57.d even 2 1 inner 1368.2.f.d yes 8
76.d even 2 1 2736.2.f.g 8
228.b odd 2 1 2736.2.f.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.f.c 8 3.b odd 2 1
1368.2.f.c 8 19.b odd 2 1
1368.2.f.d yes 8 1.a even 1 1 trivial
1368.2.f.d yes 8 57.d even 2 1 inner
2736.2.f.g 8 12.b even 2 1
2736.2.f.g 8 76.d even 2 1
2736.2.f.h 8 4.b odd 2 1
2736.2.f.h 8 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

\( T_{5}^{8} + 24T_{5}^{6} + 181T_{5}^{4} + 436T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{29}^{4} - 16T_{29}^{3} + 60T_{29}^{2} + 64T_{29} - 320 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T^{4} - 15 T^{2} - 16 T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 32 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} + 88 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{8} + 24 T^{6} + \cdots + 100 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 64 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{4} - 16 T^{3} + \cdots - 320)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 216 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$37$ \( T^{8} + 184 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 3200)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 14 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 112 T^{6} + \cdots + 58564 \) Copy content Toggle raw display
$53$ \( (T^{4} - 4 T^{3} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + \cdots + 3328)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 4976)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 208 T^{6} + \cdots + 409600 \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 280 T^{6} + \cdots + 15745024 \) Copy content Toggle raw display
$83$ \( T^{8} + 336 T^{6} + \cdots + 512656 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} + \cdots + 5888)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 272 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
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