Properties

Label 78.4.a.d
Level $78$
Weight $4$
Character orbit 78.a
Self dual yes
Analytic conductor $4.602$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 20 q^{5} - 6 q^{6} - 32 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 20 q^{5} - 6 q^{6} - 32 q^{7} + 8 q^{8} + 9 q^{9} - 40 q^{10} + 50 q^{11} - 12 q^{12} - 13 q^{13} - 64 q^{14} + 60 q^{15} + 16 q^{16} - 30 q^{17} + 18 q^{18} - 120 q^{19} - 80 q^{20} + 96 q^{21} + 100 q^{22} - 20 q^{23} - 24 q^{24} + 275 q^{25} - 26 q^{26} - 27 q^{27} - 128 q^{28} + 82 q^{29} + 120 q^{30} - 44 q^{31} + 32 q^{32} - 150 q^{33} - 60 q^{34} + 640 q^{35} + 36 q^{36} - 306 q^{37} - 240 q^{38} + 39 q^{39} - 160 q^{40} + 108 q^{41} + 192 q^{42} - 356 q^{43} + 200 q^{44} - 180 q^{45} - 40 q^{46} - 178 q^{47} - 48 q^{48} + 681 q^{49} + 550 q^{50} + 90 q^{51} - 52 q^{52} + 198 q^{53} - 54 q^{54} - 1000 q^{55} - 256 q^{56} + 360 q^{57} + 164 q^{58} + 94 q^{59} + 240 q^{60} - 62 q^{61} - 88 q^{62} - 288 q^{63} + 64 q^{64} + 260 q^{65} - 300 q^{66} - 140 q^{67} - 120 q^{68} + 60 q^{69} + 1280 q^{70} - 778 q^{71} + 72 q^{72} + 62 q^{73} - 612 q^{74} - 825 q^{75} - 480 q^{76} - 1600 q^{77} + 78 q^{78} - 1096 q^{79} - 320 q^{80} + 81 q^{81} + 216 q^{82} - 462 q^{83} + 384 q^{84} + 600 q^{85} - 712 q^{86} - 246 q^{87} + 400 q^{88} + 1224 q^{89} - 360 q^{90} + 416 q^{91} - 80 q^{92} + 132 q^{93} - 356 q^{94} + 2400 q^{95} - 96 q^{96} + 614 q^{97} + 1362 q^{98} + 450 q^{99}+O(q^{100}) \) 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 −20.0000 −6.00000 −32.0000 8.00000 9.00000 −40.0000
\(n\): 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.d 1
3.b odd 2 1 234.4.a.f 1
4.b odd 2 1 624.4.a.e 1
5.b even 2 1 1950.4.a.h 1
8.b even 2 1 2496.4.a.r 1
8.d odd 2 1 2496.4.a.i 1
12.b even 2 1 1872.4.a.r 1
13.b even 2 1 1014.4.a.d 1
13.d odd 4 2 1014.4.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.d 1 1.a even 1 1 trivial
234.4.a.f 1 3.b odd 2 1
624.4.a.e 1 4.b odd 2 1
1014.4.a.d 1 13.b even 2 1
1014.4.b.e 2 13.d odd 4 2
1872.4.a.r 1 12.b even 2 1
1950.4.a.h 1 5.b even 2 1
2496.4.a.i 1 8.d odd 2 1
2496.4.a.r 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(78))\):

\( T_{5} + 20 \) Copy content Toggle raw display
\( T_{7} + 32 \) 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
$5$ \( T + 20 \) Copy content Toggle raw display
$7$ \( T + 32 \) Copy content Toggle raw display
$11$ \( T - 50 \) Copy content Toggle raw display
$13$ \( T + 13 \) Copy content Toggle raw display
$17$ \( T + 30 \) Copy content Toggle raw display
$19$ \( T + 120 \) Copy content Toggle raw display
$23$ \( T + 20 \) Copy content Toggle raw display
$29$ \( T - 82 \) Copy content Toggle raw display
$31$ \( T + 44 \) Copy content Toggle raw display
$37$ \( T + 306 \) Copy content Toggle raw display
$41$ \( T - 108 \) Copy content Toggle raw display
$43$ \( T + 356 \) Copy content Toggle raw display
$47$ \( T + 178 \) Copy content Toggle raw display
$53$ \( T - 198 \) Copy content Toggle raw display
$59$ \( T - 94 \) Copy content Toggle raw display
$61$ \( T + 62 \) Copy content Toggle raw display
$67$ \( T + 140 \) Copy content Toggle raw display
$71$ \( T + 778 \) Copy content Toggle raw display
$73$ \( T - 62 \) Copy content Toggle raw display
$79$ \( T + 1096 \) Copy content Toggle raw display
$83$ \( T + 462 \) Copy content Toggle raw display
$89$ \( T - 1224 \) Copy content Toggle raw display
$97$ \( T - 614 \) Copy content Toggle raw display
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