Properties

Label 78.4.a.e
Level $78$
Weight $4$
Character orbit 78.a
Self dual yes
Analytic conductor $4.602$
Analytic rank $0$
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] = [78,4,Mod(1,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("78.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 78.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: \(4.60214898045\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 20 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 20 q^{7} + 8 q^{8} + 9 q^{9} + 12 q^{10} + 24 q^{11} - 12 q^{12} + 13 q^{13} + 40 q^{14} - 18 q^{15} + 16 q^{16} - 30 q^{17} + 18 q^{18} - 16 q^{19} + 24 q^{20} - 60 q^{21} + 48 q^{22} - 72 q^{23} - 24 q^{24} - 89 q^{25} + 26 q^{26} - 27 q^{27} + 80 q^{28} - 282 q^{29} - 36 q^{30} + 164 q^{31} + 32 q^{32} - 72 q^{33} - 60 q^{34} + 120 q^{35} + 36 q^{36} + 110 q^{37} - 32 q^{38} - 39 q^{39} + 48 q^{40} - 126 q^{41} - 120 q^{42} + 164 q^{43} + 96 q^{44} + 54 q^{45} - 144 q^{46} - 204 q^{47} - 48 q^{48} + 57 q^{49} - 178 q^{50} + 90 q^{51} + 52 q^{52} - 738 q^{53} - 54 q^{54} + 144 q^{55} + 160 q^{56} + 48 q^{57} - 564 q^{58} + 120 q^{59} - 72 q^{60} + 614 q^{61} + 328 q^{62} + 180 q^{63} + 64 q^{64} + 78 q^{65} - 144 q^{66} + 848 q^{67} - 120 q^{68} + 216 q^{69} + 240 q^{70} + 132 q^{71} + 72 q^{72} + 218 q^{73} + 220 q^{74} + 267 q^{75} - 64 q^{76} + 480 q^{77} - 78 q^{78} - 1096 q^{79} + 96 q^{80} + 81 q^{81} - 252 q^{82} + 552 q^{83} - 240 q^{84} - 180 q^{85} + 328 q^{86} + 846 q^{87} + 192 q^{88} + 210 q^{89} + 108 q^{90} + 260 q^{91} - 288 q^{92} - 492 q^{93} - 408 q^{94} - 96 q^{95} - 96 q^{96} - 1726 q^{97} + 114 q^{98} + 216 q^{99}+O(q^{100}) \) 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 6.00000 −6.00000 20.0000 8.00000 9.00000 12.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(13\) \( -1 \)

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 78.4.a.e 1
3.b odd 2 1 234.4.a.b 1
4.b odd 2 1 624.4.a.i 1
5.b even 2 1 1950.4.a.c 1
8.b even 2 1 2496.4.a.k 1
8.d odd 2 1 2496.4.a.b 1
12.b even 2 1 1872.4.a.e 1
13.b even 2 1 1014.4.a.b 1
13.d odd 4 2 1014.4.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.e 1 1.a even 1 1 trivial
234.4.a.b 1 3.b odd 2 1
624.4.a.i 1 4.b odd 2 1
1014.4.a.b 1 13.b even 2 1
1014.4.b.c 2 13.d odd 4 2
1872.4.a.e 1 12.b even 2 1
1950.4.a.c 1 5.b even 2 1
2496.4.a.b 1 8.d odd 2 1
2496.4.a.k 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(78))\):

\( T_{5} - 6 \) Copy content Toggle raw display
\( T_{7} - 20 \) 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
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T - 20 \) Copy content Toggle raw display
$11$ \( T - 24 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T + 30 \) Copy content Toggle raw display
$19$ \( T + 16 \) Copy content Toggle raw display
$23$ \( T + 72 \) Copy content Toggle raw display
$29$ \( T + 282 \) Copy content Toggle raw display
$31$ \( T - 164 \) Copy content Toggle raw display
$37$ \( T - 110 \) Copy content Toggle raw display
$41$ \( T + 126 \) Copy content Toggle raw display
$43$ \( T - 164 \) Copy content Toggle raw display
$47$ \( T + 204 \) Copy content Toggle raw display
$53$ \( T + 738 \) Copy content Toggle raw display
$59$ \( T - 120 \) Copy content Toggle raw display
$61$ \( T - 614 \) Copy content Toggle raw display
$67$ \( T - 848 \) Copy content Toggle raw display
$71$ \( T - 132 \) Copy content Toggle raw display
$73$ \( T - 218 \) Copy content Toggle raw display
$79$ \( T + 1096 \) Copy content Toggle raw display
$83$ \( T - 552 \) Copy content Toggle raw display
$89$ \( T - 210 \) Copy content Toggle raw display
$97$ \( T + 1726 \) Copy content Toggle raw display
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