Properties

Label 1104.2.e.f
Level $1104$
Weight $2$
Character orbit 1104.e
Analytic conductor $8.815$
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] = [1104,2,Mod(47,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1104.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1104.e (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: \(8.81548438315\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3814238552064.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 4x^{5} - 12x^{3} + 9x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} - \beta_1 q^{5} - \beta_1 q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} - \beta_1 q^{5} - \beta_1 q^{7} + \beta_{2} q^{9} + ( - \beta_{6} + \beta_{2}) q^{11} + ( - \beta_{7} + \beta_{6} - \beta_{3} - 1) q^{13} + ( - \beta_{3} + \beta_1 + 1) q^{15} + (2 \beta_{7} + \beta_{6} + \beta_{4}) q^{17} + (\beta_{4} - \beta_{2}) q^{19} + ( - \beta_{3} + \beta_1 + 1) q^{21} - q^{23} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{25} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_1 + 2) q^{27} + (2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2}) q^{29} + ( - \beta_{6} + \beta_{5} - \beta_{4}) q^{31} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{33} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 7) q^{35} + ( - \beta_{6} + \beta_{2} + 2) q^{37} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_1 + 3) q^{39} + (2 \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2}) q^{41} + ( - 2 \beta_{7} - \beta_{6} - \beta_{2} - \beta_1) q^{43} + (2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{45} + (\beta_{7} + \beta_{3} - \beta_{2} - 1) q^{47} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}) q^{49} + ( - \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 5) q^{51} + (\beta_{4} - \beta_{2}) q^{53} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{55} + (\beta_{6} + 2 \beta_{4} - \beta_{2} - \beta_1) q^{57} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 3) q^{59} + ( - 2 \beta_{6} + 2 \beta_{2} + 2) q^{61} + (2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{63} + (4 \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{2} + 3 \beta_1) q^{65} + (2 \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{67} + \beta_{7} q^{69} + (\beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - 4) q^{71} + ( - 2 \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} - 2) q^{73} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 7) q^{75} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{77} + ( - \beta_{4} + \beta_{2} - 2 \beta_1) q^{79} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_1) q^{81} + (2 \beta_{7} - \beta_{6} + 2 \beta_{3} - \beta_{2} - 2) q^{83} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 3) q^{85} + ( - 3 \beta_{4} - \beta_{3} + \beta_1 + 4) q^{87} + (2 \beta_{7} + \beta_{6} + \beta_{2} + 3 \beta_1) q^{89} + (4 \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{2} + 3 \beta_1) q^{91} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{93} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{95} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{3} + \beta_{2}) q^{97} + (\beta_{6} - \beta_{4} - \beta_{3} - 3 \beta_1 + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{9} - 4 q^{11} - 4 q^{13} + 10 q^{15} + 10 q^{21} - 8 q^{23} - 16 q^{25} + 12 q^{27} - 10 q^{33} - 56 q^{35} + 12 q^{37} + 24 q^{39} - 6 q^{45} - 8 q^{47} + 38 q^{51} - 16 q^{59} + 8 q^{61} - 6 q^{63} - 24 q^{71} - 20 q^{73} + 52 q^{75} + 2 q^{81} - 20 q^{83} - 24 q^{85} + 40 q^{87} + 2 q^{93} + 8 q^{95} + 4 q^{97} + 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} - 4x^{5} - 12x^{3} + 9x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 8\nu^{5} + 20\nu^{4} - 24\nu^{3} + 24\nu^{2} - 63\nu + 81 ) / 108 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - \nu^{4} + 4\nu^{3} + 12\nu - 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 10\nu^{5} + 4\nu^{4} + 18\nu^{3} - 6\nu^{2} - 9\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - \nu^{6} + 4\nu^{5} + 4\nu^{4} + 16\nu^{3} + 12\nu^{2} + 21\nu - 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 4\nu^{5} - 4\nu^{4} - 16\nu^{3} + 24\nu^{2} + 51\nu + 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{5} + 4\nu^{4} - 15\nu^{2} + 18\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + \nu^{5} - 4\nu^{4} - 12\nu^{2} + 9\nu ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} + 3\beta_{3} - 3\beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 2\beta_{4} - \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{7} + 2\beta_{6} - 4\beta_{5} + 2\beta_{4} - 12\beta_{3} - 6\beta _1 + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{7} + 2\beta_{6} + 8\beta_{5} + 14\beta_{4} + 12\beta_{3} - 24\beta_{2} - 18\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 55\beta_{7} - 11\beta_{6} + 7\beta_{5} + 25\beta_{4} + 12\beta_{3} - 12\beta_{2} + 30\beta _1 - 12 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(277\) \(415\) \(737\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−1.42459 + 0.985156i
−1.42459 0.985156i
−0.696260 + 1.58595i
−0.696260 1.58595i
0.453491 + 1.67163i
0.453491 1.67163i
1.66736 + 0.468935i
1.66736 0.468935i
0 −1.42459 0.985156i 0 4.29054i 0 4.29054i 0 1.05894 + 2.80689i 0
47.2 0 −1.42459 + 0.985156i 0 4.29054i 0 4.29054i 0 1.05894 2.80689i 0
47.3 0 −0.696260 1.58595i 0 2.04950i 0 2.04950i 0 −2.03044 + 2.20846i 0
47.4 0 −0.696260 + 1.58595i 0 2.04950i 0 2.04950i 0 −2.03044 2.20846i 0
47.5 0 0.453491 1.67163i 0 2.19318i 0 2.19318i 0 −2.58869 1.51614i 0
47.6 0 0.453491 + 1.67163i 0 2.19318i 0 2.19318i 0 −2.58869 + 1.51614i 0
47.7 0 1.66736 0.468935i 0 0.762065i 0 0.762065i 0 2.56020 1.56377i 0
47.8 0 1.66736 + 0.468935i 0 0.762065i 0 0.762065i 0 2.56020 + 1.56377i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.2.e.f 8
3.b odd 2 1 1104.2.e.g yes 8
4.b odd 2 1 1104.2.e.g yes 8
12.b even 2 1 inner 1104.2.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1104.2.e.f 8 1.a even 1 1 trivial
1104.2.e.f 8 12.b even 2 1 inner
1104.2.e.g yes 8 3.b odd 2 1
1104.2.e.g yes 8 4.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 \(S_{2}^{\mathrm{new}}(1104, [\chi])\):

\( T_{5}^{8} + 28T_{5}^{6} + 202T_{5}^{4} + 480T_{5}^{2} + 216 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 34T_{11}^{2} - 32T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{6} - 4 T^{5} - 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 28 T^{6} + 202 T^{4} + \cdots + 216 \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + 202 T^{4} + \cdots + 216 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 34 T^{2} - 32 T + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 33 T^{2} - 68 T + 74)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 92 T^{6} + 2682 T^{4} + \cdots + 17496 \) Copy content Toggle raw display
$19$ \( T^{8} + 60 T^{6} + 486 T^{4} + \cdots + 96 \) Copy content Toggle raw display
$23$ \( (T + 1)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 190 T^{6} + 11425 T^{4} + \cdots + 1452384 \) Copy content Toggle raw display
$31$ \( T^{8} + 146 T^{6} + 6849 T^{4} + \cdots + 629856 \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} - 22 T^{2} + 96 T + 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 126 T^{6} + 3153 T^{4} + \cdots + 24576 \) Copy content Toggle raw display
$43$ \( T^{8} + 116 T^{6} + 2310 T^{4} + \cdots + 31104 \) Copy content Toggle raw display
$47$ \( (T^{4} + 4 T^{3} - 29 T^{2} - 156 T - 162)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 60 T^{6} + 486 T^{4} + \cdots + 96 \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} - 130 T^{2} - 344 T + 2832)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} - 136 T^{2} + 304 T + 2256)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 300 T^{6} + 25434 T^{4} + \cdots + 1417176 \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} - 39 T^{2} - 560 T + 192)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 10 T^{3} - 67 T^{2} - 976 T - 2592)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 188 T^{6} + 8358 T^{4} + \cdots + 629856 \) Copy content Toggle raw display
$83$ \( (T^{4} + 10 T^{3} - 66 T^{2} - 864 T - 1728)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 276 T^{6} + 18918 T^{4} + \cdots + 4153344 \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} - 102 T^{2} + 512 T - 328)^{2} \) Copy content Toggle raw display
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