Properties

Label 115.4.a.b
Level $115$
Weight $4$
Character orbit 115.a
Self dual yes
Analytic conductor $6.785$
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] = [115,4,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, 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("115.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(6.78521965066\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 5 q^{5} - 6 q^{6} - 2 q^{7} - 24 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 5 q^{5} - 6 q^{6} - 2 q^{7} - 24 q^{8} - 18 q^{9} + 10 q^{10} - 16 q^{11} + 12 q^{12} - 47 q^{13} - 4 q^{14} - 15 q^{15} - 16 q^{16} - 24 q^{17} - 36 q^{18} - 56 q^{19} - 20 q^{20} + 6 q^{21} - 32 q^{22} - 23 q^{23} + 72 q^{24} + 25 q^{25} - 94 q^{26} + 135 q^{27} + 8 q^{28} + 85 q^{29} - 30 q^{30} + 67 q^{31} + 160 q^{32} + 48 q^{33} - 48 q^{34} - 10 q^{35} + 72 q^{36} + 104 q^{37} - 112 q^{38} + 141 q^{39} - 120 q^{40} - 53 q^{41} + 12 q^{42} - 234 q^{43} + 64 q^{44} - 90 q^{45} - 46 q^{46} + 285 q^{47} + 48 q^{48} - 339 q^{49} + 50 q^{50} + 72 q^{51} + 188 q^{52} + 2 q^{53} + 270 q^{54} - 80 q^{55} + 48 q^{56} + 168 q^{57} + 170 q^{58} + 80 q^{59} + 60 q^{60} - 764 q^{61} + 134 q^{62} + 36 q^{63} + 448 q^{64} - 235 q^{65} + 96 q^{66} + 236 q^{67} + 96 q^{68} + 69 q^{69} - 20 q^{70} - 289 q^{71} + 432 q^{72} - 225 q^{73} + 208 q^{74} - 75 q^{75} + 224 q^{76} + 32 q^{77} + 282 q^{78} + 24 q^{79} - 80 q^{80} + 81 q^{81} - 106 q^{82} + 684 q^{83} - 24 q^{84} - 120 q^{85} - 468 q^{86} - 255 q^{87} + 384 q^{88} - 1370 q^{89} - 180 q^{90} + 94 q^{91} + 92 q^{92} - 201 q^{93} + 570 q^{94} - 280 q^{95} - 480 q^{96} - 110 q^{97} - 678 q^{98} + 288 q^{99}+O(q^{100}) \) 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 5.00000 −6.00000 −2.00000 −24.0000 −18.0000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(23\) \(1\)

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 115.4.a.b 1
3.b odd 2 1 1035.4.a.a 1
4.b odd 2 1 1840.4.a.f 1
5.b even 2 1 575.4.a.a 1
5.c odd 4 2 575.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.b 1 1.a even 1 1 trivial
575.4.a.a 1 5.b even 2 1
575.4.b.a 2 5.c odd 4 2
1035.4.a.a 1 3.b odd 2 1
1840.4.a.f 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\). 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
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 16 \) Copy content Toggle raw display
$13$ \( T + 47 \) Copy content Toggle raw display
$17$ \( T + 24 \) Copy content Toggle raw display
$19$ \( T + 56 \) Copy content Toggle raw display
$23$ \( T + 23 \) Copy content Toggle raw display
$29$ \( T - 85 \) Copy content Toggle raw display
$31$ \( T - 67 \) Copy content Toggle raw display
$37$ \( T - 104 \) Copy content Toggle raw display
$41$ \( T + 53 \) Copy content Toggle raw display
$43$ \( T + 234 \) Copy content Toggle raw display
$47$ \( T - 285 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T - 80 \) Copy content Toggle raw display
$61$ \( T + 764 \) Copy content Toggle raw display
$67$ \( T - 236 \) Copy content Toggle raw display
$71$ \( T + 289 \) Copy content Toggle raw display
$73$ \( T + 225 \) Copy content Toggle raw display
$79$ \( T - 24 \) Copy content Toggle raw display
$83$ \( T - 684 \) Copy content Toggle raw display
$89$ \( T + 1370 \) Copy content Toggle raw display
$97$ \( T + 110 \) Copy content Toggle raw display
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