Properties

Label 798.4.a.c
Level $798$
Weight $4$
Character orbit 798.a
Self dual yes
Analytic conductor $47.084$
Analytic rank $1$
Dimension $1$
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] = [798,4,Mod(1,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 798.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: \(47.0835241846\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} - 42 q^{11} - 12 q^{12} + 20 q^{13} + 14 q^{14} + 16 q^{16} - 96 q^{17} + 18 q^{18} + 19 q^{19} - 21 q^{21} - 84 q^{22} + 102 q^{23} - 24 q^{24} - 125 q^{25} + 40 q^{26} - 27 q^{27} + 28 q^{28} + 90 q^{29} - 196 q^{31} + 32 q^{32} + 126 q^{33} - 192 q^{34} + 36 q^{36} - 214 q^{37} + 38 q^{38} - 60 q^{39} + 378 q^{41} - 42 q^{42} - 376 q^{43} - 168 q^{44} + 204 q^{46} + 216 q^{47} - 48 q^{48} + 49 q^{49} - 250 q^{50} + 288 q^{51} + 80 q^{52} - 750 q^{53} - 54 q^{54} + 56 q^{56} - 57 q^{57} + 180 q^{58} - 252 q^{59} + 182 q^{61} - 392 q^{62} + 63 q^{63} + 64 q^{64} + 252 q^{66} - 286 q^{67} - 384 q^{68} - 306 q^{69} + 264 q^{71} + 72 q^{72} - 358 q^{73} - 428 q^{74} + 375 q^{75} + 76 q^{76} - 294 q^{77} - 120 q^{78} - 862 q^{79} + 81 q^{81} + 756 q^{82} - 384 q^{83} - 84 q^{84} - 752 q^{86} - 270 q^{87} - 336 q^{88} + 42 q^{89} + 140 q^{91} + 408 q^{92} + 588 q^{93} + 432 q^{94} - 96 q^{96} - 1240 q^{97} + 98 q^{98} - 378 q^{99}+O(q^{100}) \) 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
0
2.00000 −3.00000 4.00000 0 −6.00000 7.00000 8.00000 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(19\) \(-1\)

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 798.4.a.c 1
3.b odd 2 1 2394.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.4.a.c 1 1.a even 1 1 trivial
2394.4.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(798))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 42 \) Copy content Toggle raw display
$13$ \( T - 20 \) Copy content Toggle raw display
$17$ \( T + 96 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T - 102 \) Copy content Toggle raw display
$29$ \( T - 90 \) Copy content Toggle raw display
$31$ \( T + 196 \) Copy content Toggle raw display
$37$ \( T + 214 \) Copy content Toggle raw display
$41$ \( T - 378 \) Copy content Toggle raw display
$43$ \( T + 376 \) Copy content Toggle raw display
$47$ \( T - 216 \) Copy content Toggle raw display
$53$ \( T + 750 \) Copy content Toggle raw display
$59$ \( T + 252 \) Copy content Toggle raw display
$61$ \( T - 182 \) Copy content Toggle raw display
$67$ \( T + 286 \) Copy content Toggle raw display
$71$ \( T - 264 \) Copy content Toggle raw display
$73$ \( T + 358 \) Copy content Toggle raw display
$79$ \( T + 862 \) Copy content Toggle raw display
$83$ \( T + 384 \) Copy content Toggle raw display
$89$ \( T - 42 \) Copy content Toggle raw display
$97$ \( T + 1240 \) Copy content Toggle raw display
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