Properties

Label 799.2.b.a
Level $799$
Weight $2$
Character orbit 799.b
Analytic conductor $6.380$
Analytic rank $0$
Dimension $2$
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] = [799,2,Mod(424,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("799.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 799.b (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: \(6.38004712150\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} - q^{4} - \beta q^{5} + \beta q^{6} - \beta q^{7} - 3 q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} - q^{4} - \beta q^{5} + \beta q^{6} - \beta q^{7} - 3 q^{8} - q^{9} - \beta q^{10} - \beta q^{12} - 6 q^{13} - \beta q^{14} + 4 q^{15} - q^{16} + ( - 2 \beta + 1) q^{17} - q^{18} - 8 q^{19} + \beta q^{20} + 4 q^{21} - 4 \beta q^{23} - 3 \beta q^{24} + q^{25} - 6 q^{26} + 2 \beta q^{27} + \beta q^{28} - 3 \beta q^{29} + 4 q^{30} + 2 \beta q^{31} + 5 q^{32} + ( - 2 \beta + 1) q^{34} - 4 q^{35} + q^{36} - 8 q^{38} - 6 \beta q^{39} + 3 \beta q^{40} + \beta q^{41} + 4 q^{42} + \beta q^{45} - 4 \beta q^{46} - q^{47} - \beta q^{48} + 3 q^{49} + q^{50} + (\beta + 8) q^{51} + 6 q^{52} + 6 q^{53} + 2 \beta q^{54} + 3 \beta q^{56} - 8 \beta q^{57} - 3 \beta q^{58} - 12 q^{59} - 4 q^{60} + 4 \beta q^{61} + 2 \beta q^{62} + \beta q^{63} + 7 q^{64} + 6 \beta q^{65} - 8 q^{67} + (2 \beta - 1) q^{68} + 16 q^{69} - 4 q^{70} - 5 \beta q^{71} + 3 q^{72} + \beta q^{73} + \beta q^{75} + 8 q^{76} - 6 \beta q^{78} - 7 \beta q^{79} + \beta q^{80} - 11 q^{81} + \beta q^{82} - 4 q^{83} - 4 q^{84} + ( - \beta - 8) q^{85} + 12 q^{87} + 10 q^{89} + \beta q^{90} + 6 \beta q^{91} + 4 \beta q^{92} - 8 q^{93} - q^{94} + 8 \beta q^{95} + 5 \beta q^{96} - 4 \beta q^{97} + 3 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} - 12 q^{13} + 8 q^{15} - 2 q^{16} + 2 q^{17} - 2 q^{18} - 16 q^{19} + 8 q^{21} + 2 q^{25} - 12 q^{26} + 8 q^{30} + 10 q^{32} + 2 q^{34} - 8 q^{35} + 2 q^{36} - 16 q^{38} + 8 q^{42} - 2 q^{47} + 6 q^{49} + 2 q^{50} + 16 q^{51} + 12 q^{52} + 12 q^{53} - 24 q^{59} - 8 q^{60} + 14 q^{64} - 16 q^{67} - 2 q^{68} + 32 q^{69} - 8 q^{70} + 6 q^{72} + 16 q^{76} - 22 q^{81} - 8 q^{83} - 8 q^{84} - 16 q^{85} + 24 q^{87} + 20 q^{89} - 16 q^{93} - 2 q^{94} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
1.00000i
1.00000i
1.00000 2.00000i −1.00000 2.00000i 2.00000i 2.00000i −3.00000 −1.00000 2.00000i
424.2 1.00000 2.00000i −1.00000 2.00000i 2.00000i 2.00000i −3.00000 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.2.b.a 2
17.b even 2 1 inner 799.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.b.a 2 1.a even 1 1 trivial
799.2.b.a 2 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 17 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 100 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 196 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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