Properties

Label 804.2.c.a
Level $804$
Weight $2$
Character orbit 804.c
Analytic conductor $6.420$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [804,2,Mod(671,804)]
 
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([1, 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("804.671");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.c (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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} - 2 q^{4} - \beta_{2} q^{5} + ( - \beta_{3} + 2 \beta_1) q^{6} - 2 \beta_1 q^{7} - 2 \beta_{2} q^{8} + (2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} - 2 q^{4} - \beta_{2} q^{5} + ( - \beta_{3} + 2 \beta_1) q^{6} - 2 \beta_1 q^{7} - 2 \beta_{2} q^{8} + (2 \beta_{2} + 1) q^{9} + 2 q^{10} + 4 \beta_{3} q^{11} + ( - 2 \beta_{3} - 2 \beta_1) q^{12} + 2 q^{13} + 2 \beta_{3} q^{14} + (\beta_{3} - 2 \beta_1) q^{15} + 4 q^{16} - 4 \beta_{2} q^{17} + (\beta_{2} - 4) q^{18} - 4 \beta_1 q^{19} + 2 \beta_{2} q^{20} + ( - 2 \beta_{2} + 2) q^{21} + 8 \beta_1 q^{22} - 5 \beta_{3} q^{23} + (2 \beta_{3} - 4 \beta_1) q^{24} + 3 q^{25} + 2 \beta_{2} q^{26} + ( - \beta_{3} + 5 \beta_1) q^{27} + 4 \beta_1 q^{28} + 4 \beta_{2} q^{29} + (2 \beta_{3} + 2 \beta_1) q^{30} + 4 \beta_{2} q^{32} + (4 \beta_{2} + 8) q^{33} + 8 q^{34} - 2 \beta_{3} q^{35} + ( - 4 \beta_{2} - 2) q^{36} + 8 q^{37} + 4 \beta_{3} q^{38} + (2 \beta_{3} + 2 \beta_1) q^{39} - 4 q^{40} + \beta_{2} q^{41} + (2 \beta_{2} + 4) q^{42} + 4 \beta_1 q^{43} - 8 \beta_{3} q^{44} + ( - \beta_{2} + 4) q^{45} - 10 \beta_1 q^{46} - 5 \beta_{3} q^{47} + (4 \beta_{3} + 4 \beta_1) q^{48} + 3 q^{49} + 3 \beta_{2} q^{50} + (4 \beta_{3} - 8 \beta_1) q^{51} - 4 q^{52} + 9 \beta_{2} q^{53} + ( - 5 \beta_{3} - 2 \beta_1) q^{54} - 8 \beta_1 q^{55} - 4 \beta_{3} q^{56} + ( - 4 \beta_{2} + 4) q^{57} - 8 q^{58} - \beta_{3} q^{59} + ( - 2 \beta_{3} + 4 \beta_1) q^{60} - 6 q^{61} + (4 \beta_{3} - 2 \beta_1) q^{63} - 8 q^{64} - 2 \beta_{2} q^{65} + (8 \beta_{2} - 8) q^{66} - \beta_1 q^{67} + 8 \beta_{2} q^{68} + ( - 5 \beta_{2} - 10) q^{69} - 4 \beta_1 q^{70} - 11 \beta_{3} q^{71} + ( - 2 \beta_{2} + 8) q^{72} + 10 q^{73} + 8 \beta_{2} q^{74} + (3 \beta_{3} + 3 \beta_1) q^{75} + 8 \beta_1 q^{76} - 8 \beta_{2} q^{77} + ( - 2 \beta_{3} + 4 \beta_1) q^{78} - 8 \beta_1 q^{79} - 4 \beta_{2} q^{80} + (4 \beta_{2} - 7) q^{81} - 2 q^{82} + 11 \beta_{3} q^{83} + (4 \beta_{2} - 4) q^{84} - 8 q^{85} - 4 \beta_{3} q^{86} + ( - 4 \beta_{3} + 8 \beta_1) q^{87} - 16 \beta_1 q^{88} - 12 \beta_{2} q^{89} + (4 \beta_{2} + 2) q^{90} - 4 \beta_1 q^{91} + 10 \beta_{3} q^{92} - 10 \beta_1 q^{94} - 4 \beta_{3} q^{95} + ( - 4 \beta_{3} + 8 \beta_1) q^{96} - 6 q^{97} + 3 \beta_{2} q^{98} + (4 \beta_{3} + 16 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 4 q^{9} + 8 q^{10} + 8 q^{13} + 16 q^{16} - 16 q^{18} + 8 q^{21} + 12 q^{25} + 32 q^{33} + 32 q^{34} - 8 q^{36} + 32 q^{37} - 16 q^{40} + 16 q^{42} + 16 q^{45} + 12 q^{49} - 16 q^{52} + 16 q^{57} - 32 q^{58} - 24 q^{61} - 32 q^{64} - 32 q^{66} - 40 q^{69} + 32 q^{72} + 40 q^{73} - 28 q^{81} - 8 q^{82} - 16 q^{84} - 32 q^{85} + 8 q^{90} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
1.41421i −1.41421 + 1.00000i −2.00000 1.41421i 1.41421 + 2.00000i 2.00000i 2.82843i 1.00000 2.82843i 2.00000
671.2 1.41421i 1.41421 1.00000i −2.00000 1.41421i −1.41421 2.00000i 2.00000i 2.82843i 1.00000 2.82843i 2.00000
671.3 1.41421i −1.41421 1.00000i −2.00000 1.41421i 1.41421 2.00000i 2.00000i 2.82843i 1.00000 + 2.82843i 2.00000
671.4 1.41421i 1.41421 + 1.00000i −2.00000 1.41421i −1.41421 + 2.00000i 2.00000i 2.82843i 1.00000 + 2.82843i 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
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 804.2.c.a 4
3.b odd 2 1 inner 804.2.c.a 4
4.b odd 2 1 inner 804.2.c.a 4
12.b even 2 1 inner 804.2.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.c.a 4 1.a even 1 1 trivial
804.2.c.a 4 3.b odd 2 1 inner
804.2.c.a 4 4.b odd 2 1 inner
804.2.c.a 4 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T - 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 242)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 242)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{4} \) Copy content Toggle raw display
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