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Properties

Label 1104.3.k.a
Level $1104$
Weight $3$
Character orbit 1104.k
Analytic conductor $30.082$
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] = [1104,3,Mod(415,1104)]
 
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, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1104.k (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: \(30.0818211854\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3830743449.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 3x^{5} + x^{4} - 9x^{3} + 36x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( - \beta_{3} - 2) q^{5} + (2 \beta_{4} + \beta_{2}) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + ( - \beta_{3} - 2) q^{5} + (2 \beta_{4} + \beta_{2}) q^{7} - 3 q^{9} + (3 \beta_{6} + 3 \beta_{4} - \beta_{2}) q^{11} + (\beta_{7} + \beta_{5} + 3 \beta_{3} - 1) q^{13} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{15} + (\beta_{5} + \beta_{3} + 13) q^{17} + (3 \beta_{6} - 3 \beta_{4} + 4 \beta_1) q^{19} + ( - \beta_{5} - \beta_{3} - 7) q^{21} + \beta_{6} q^{23} + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots + 1) q^{25}+ \cdots + ( - 9 \beta_{6} - 9 \beta_{4} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{5} - 24 q^{9} + 108 q^{17} - 60 q^{21} + 8 q^{25} + 80 q^{29} - 60 q^{33} - 172 q^{37} - 80 q^{41} + 48 q^{45} + 48 q^{49} + 232 q^{53} + 24 q^{57} + 300 q^{61} - 464 q^{65} + 216 q^{73} - 232 q^{77} + 72 q^{81} - 360 q^{85} - 244 q^{89} - 96 q^{93} + 560 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 3x^{5} + x^{4} - 9x^{3} + 36x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} - 216\nu^{6} + 116\nu^{5} - 306\nu^{4} + 56\nu^{3} + 42\nu^{2} + 3978\nu - 3537 ) / 891 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} + 72\nu^{6} + 130\nu^{5} - 162\nu^{4} + 370\nu^{3} - 960\nu^{2} + 126\nu - 108 ) / 891 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + 8\nu^{5} - 36\nu^{4} + 2\nu^{3} + 42\nu^{2} - 72\nu - 54 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52\nu^{7} - 72\nu^{6} + 46\nu^{5} - 36\nu^{4} + 268\nu^{3} - 492\nu^{2} + 1260\nu - 1971 ) / 891 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -10\nu^{7} + 14\nu^{5} + 18\nu^{4} + 44\nu^{3} + 114\nu^{2} - 234\nu + 189 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -46\nu^{7} + 84\nu^{6} - 94\nu^{5} - 24\nu^{4} + 98\nu^{3} + 90\nu^{2} - 1404\nu + 2349 ) / 297 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -20\nu^{7} + 36\nu^{6} - 26\nu^{5} + 18\nu^{4} + 34\nu^{3} + 264\nu^{2} - 414\nu + 864 ) / 81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 9\beta_{4} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 5\beta_{6} + 3\beta_{5} + 2\beta_{4} - 11\beta_{3} - \beta_{2} - 5\beta _1 - 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{6} + 2\beta_{5} - 6\beta_{4} + \beta_{3} + 5\beta_{2} - \beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9\beta_{7} - 10\beta_{6} - 9\beta_{5} - 5\beta_{4} + 15\beta_{2} - 5\beta _1 - 25 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{7} - 11\beta_{6} - 8\beta_{5} + 36\beta_{4} + 16\beta_{3} - 10\beta_{2} - 19\beta _1 + 66 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
−1.44007 − 0.962388i
1.12321 − 1.31848i
1.70345 − 0.313488i
0.113417 + 1.72833i
−1.44007 + 0.962388i
1.12321 + 1.31848i
1.70345 + 0.313488i
0.113417 − 1.72833i
0 − 1.73205i 0 −8.21396 0 − 11.4876i 0 −3.00000 0
415.2 0 − 1.73205i 0 −4.32094 0 1.85549i 0 −3.00000 0
415.3 0 − 1.73205i 0 0.320938 0 − 5.71991i 0 −3.00000 0
415.4 0 − 1.73205i 0 4.21396 0 − 1.96849i 0 −3.00000 0
415.5 0 1.73205i 0 −8.21396 0 11.4876i 0 −3.00000 0
415.6 0 1.73205i 0 −4.32094 0 − 1.85549i 0 −3.00000 0
415.7 0 1.73205i 0 0.320938 0 5.71991i 0 −3.00000 0
415.8 0 1.73205i 0 4.21396 0 1.96849i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.3.k.a ✓ 8
4.b odd 2 1 inner 1104.3.k.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1104.3.k.a ✓ 8 1.a even 1 1 trivial
1104.3.k.a ✓ 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8T_{5}^{3} - 20T_{5}^{2} - 144T_{5} + 48 \) acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{3} - 20 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 172 T^{6} + \cdots + 57600 \) Copy content Toggle raw display
$11$ \( T^{8} + 724 T^{6} + \cdots + 116640000 \) Copy content Toggle raw display
$13$ \( (T^{4} - 632 T^{2} + 60112)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 54 T^{3} + \cdots + 10800)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 279951042816 \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 40 T^{3} + \cdots - 16368)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 52027785216 \) Copy content Toggle raw display
$37$ \( (T^{4} + 86 T^{3} + \cdots + 600048)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 40 T^{3} + \cdots - 944880)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 3358745305344 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 22428057600 \) Copy content Toggle raw display
$53$ \( (T^{4} - 116 T^{3} + \cdots - 19772880)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 1051199078400 \) Copy content Toggle raw display
$61$ \( (T^{4} - 150 T^{3} + \cdots + 106480)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 1330091663616 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 7057056006144 \) Copy content Toggle raw display
$73$ \( (T^{4} - 108 T^{3} + \cdots + 13193200)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 716690923776 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 22413482866944 \) Copy content Toggle raw display
$89$ \( (T^{4} + 122 T^{3} + \cdots + 29198640)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 280 T^{3} + \cdots - 27815600)^{2} \) Copy content Toggle raw display
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