Properties

Label 804.2.a.f
Level $804$
Weight $2$
Character orbit 804.a
Self dual yes
Analytic conductor $6.420$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [804,2,Mod(1,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.a (trivial)

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: yes
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.24571284.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 15x^{3} + 10x^{2} + 64x + 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{2} + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{2} + 1) q^{7} + q^{9} + (\beta_{4} + 1) q^{11} + ( - \beta_{4} + 1) q^{13} + ( - \beta_1 + 1) q^{15} + ( - \beta_{3} + \beta_{2}) q^{17} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{19} + ( - \beta_{2} + 1) q^{21} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{23} + (\beta_{2} + 2) q^{25} + q^{27} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{29} + (\beta_{4} - 2 \beta_{3} - \beta_{2}) q^{31} + (\beta_{4} + 1) q^{33} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{35} + (\beta_{4} - \beta_{3} + 2) q^{37} + ( - \beta_{4} + 1) q^{39} + (\beta_{4} + 2 \beta_{3} + \beta_1 + 2) q^{41} + ( - \beta_{4} + \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_1 + 1) q^{45} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{47} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + 5) q^{49} + ( - \beta_{3} + \beta_{2}) q^{51} + ( - \beta_{4} + \beta_1 - 2) q^{53} + (\beta_{4} + 2 \beta_{3} + 3) q^{55} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{57} + ( - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{59} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{61} + ( - \beta_{2} + 1) q^{63} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{65} + q^{67} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{69} + (\beta_{4} - 5) q^{71} + ( - \beta_{3} - 2 \beta_1 + 3) q^{73} + (\beta_{2} + 2) q^{75} + (3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{77} + (2 \beta_{2} - 2 \beta_1) q^{79} + q^{81} + ( - \beta_{4} - 3 \beta_1 - 4) q^{83} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 1) q^{85} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{87} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{89} + ( - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{91} + (\beta_{4} - 2 \beta_{3} - \beta_{2}) q^{93} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 5) q^{95} + ( - 3 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{97} + (\beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 6 q^{11} + 4 q^{13} + 3 q^{15} + q^{17} + 13 q^{19} + 5 q^{21} + 10 q^{25} + 5 q^{27} + 3 q^{29} + 3 q^{31} + 6 q^{33} - 9 q^{35} + 12 q^{37} + 4 q^{39} + 11 q^{41} + 3 q^{43} + 3 q^{45} - 13 q^{47} + 24 q^{49} + q^{51} - 9 q^{53} + 14 q^{55} + 13 q^{57} - 12 q^{59} + 4 q^{61} + 5 q^{63} - 8 q^{65} + 5 q^{67} - 24 q^{71} + 12 q^{73} + 10 q^{75} - 4 q^{77} - 4 q^{79} + 5 q^{81} - 27 q^{83} - 4 q^{85} + 3 q^{87} - 5 q^{89} + 14 q^{91} + 3 q^{93} - 30 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 15x^{3} + 10x^{2} + 64x + 38 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 5\nu^{3} - 8\nu^{2} + 34\nu + 34 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 10\nu^{2} + 18\nu + 28 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + \beta_{2} + 10\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{4} - 6\beta_{3} + 13\beta_{2} + 32\beta _1 + 59 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.96721
2.95721
−0.789611
−1.65253
−2.48228
0 1.00000 0 −2.96721 0 −0.804360 0 1.00000 0
1.2 0 1.00000 0 −1.95721 0 4.16931 0 1.00000 0
1.3 0 1.00000 0 1.78961 0 4.79729 0 1.00000 0
1.4 0 1.00000 0 2.65253 0 0.964067 0 1.00000 0
1.5 0 1.00000 0 3.48228 0 −4.12631 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(67\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.a.f 5
3.b odd 2 1 2412.2.a.j 5
4.b odd 2 1 3216.2.a.ba 5
12.b even 2 1 9648.2.a.ca 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.a.f 5 1.a even 1 1 trivial
2412.2.a.j 5 3.b odd 2 1
3216.2.a.ba 5 4.b odd 2 1
9648.2.a.ca 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 3T_{5}^{4} - 13T_{5}^{3} + 37T_{5}^{2} + 36T_{5} - 96 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(804))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} - 13 T^{3} + 37 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$7$ \( T^{5} - 5 T^{4} - 17 T^{3} + 89 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{5} - 6 T^{4} - 16 T^{3} + 100 T^{2} + \cdots - 384 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 24 T^{3} + 60 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$17$ \( T^{5} - T^{4} - 64 T^{3} + 44 T^{2} + \cdots - 1128 \) Copy content Toggle raw display
$19$ \( T^{5} - 13 T^{4} + 4 T^{3} + \cdots - 2896 \) Copy content Toggle raw display
$23$ \( T^{5} - 60 T^{3} + 72 T^{2} + \cdots + 162 \) Copy content Toggle raw display
$29$ \( T^{5} - 3 T^{4} - 58 T^{3} + \cdots - 1152 \) Copy content Toggle raw display
$31$ \( T^{5} - 3 T^{4} - 137 T^{3} + \cdots - 11104 \) Copy content Toggle raw display
$37$ \( T^{5} - 12 T^{4} + 8 T^{3} + 246 T^{2} + \cdots + 346 \) Copy content Toggle raw display
$41$ \( T^{5} - 11 T^{4} - 111 T^{3} + \cdots - 22368 \) Copy content Toggle raw display
$43$ \( T^{5} - 3 T^{4} - 107 T^{3} + \cdots - 1096 \) Copy content Toggle raw display
$47$ \( T^{5} + 13 T^{4} - 112 T^{3} + \cdots + 19128 \) Copy content Toggle raw display
$53$ \( T^{5} + 9 T^{4} - 25 T^{3} - 233 T^{2} + \cdots - 72 \) Copy content Toggle raw display
$59$ \( T^{5} + 12 T^{4} - 42 T^{3} - 136 T^{2} + \cdots + 18 \) Copy content Toggle raw display
$61$ \( T^{5} - 4 T^{4} - 168 T^{3} + \cdots - 9136 \) Copy content Toggle raw display
$67$ \( (T - 1)^{5} \) Copy content Toggle raw display
$71$ \( T^{5} + 24 T^{4} + 200 T^{3} + \cdots + 192 \) Copy content Toggle raw display
$73$ \( T^{5} - 12 T^{4} - 8 T^{3} + 386 T^{2} + \cdots + 362 \) Copy content Toggle raw display
$79$ \( T^{5} + 4 T^{4} - 216 T^{3} + \cdots + 16384 \) Copy content Toggle raw display
$83$ \( T^{5} + 27 T^{4} + 143 T^{3} + \cdots + 3336 \) Copy content Toggle raw display
$89$ \( T^{5} + 5 T^{4} - 130 T^{3} + \cdots - 2424 \) Copy content Toggle raw display
$97$ \( T^{5} - 8 T^{4} - 348 T^{3} + \cdots - 26912 \) Copy content Toggle raw display
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