Properties

Label 242.4.a.a
Level $242$
Weight $4$
Character orbit 242.a
Self dual yes
Analytic conductor $14.278$
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] = [242,4,Mod(1,242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(242, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.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: \(14.2784622214\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + q^{3} + 4 q^{4} - 3 q^{5} - 2 q^{6} + 10 q^{7} - 8 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + q^{3} + 4 q^{4} - 3 q^{5} - 2 q^{6} + 10 q^{7} - 8 q^{8} - 26 q^{9} + 6 q^{10} + 4 q^{12} + 16 q^{13} - 20 q^{14} - 3 q^{15} + 16 q^{16} - 42 q^{17} + 52 q^{18} - 116 q^{19} - 12 q^{20} + 10 q^{21} + 189 q^{23} - 8 q^{24} - 116 q^{25} - 32 q^{26} - 53 q^{27} + 40 q^{28} + 120 q^{29} + 6 q^{30} - 163 q^{31} - 32 q^{32} + 84 q^{34} - 30 q^{35} - 104 q^{36} - 409 q^{37} + 232 q^{38} + 16 q^{39} + 24 q^{40} - 468 q^{41} - 20 q^{42} - 110 q^{43} + 78 q^{45} - 378 q^{46} + 144 q^{47} + 16 q^{48} - 243 q^{49} + 232 q^{50} - 42 q^{51} + 64 q^{52} + 90 q^{53} + 106 q^{54} - 80 q^{56} - 116 q^{57} - 240 q^{58} - 453 q^{59} - 12 q^{60} - 20 q^{61} + 326 q^{62} - 260 q^{63} + 64 q^{64} - 48 q^{65} - 97 q^{67} - 168 q^{68} + 189 q^{69} + 60 q^{70} - 465 q^{71} + 208 q^{72} - 848 q^{73} + 818 q^{74} - 116 q^{75} - 464 q^{76} - 32 q^{78} + 742 q^{79} - 48 q^{80} + 649 q^{81} + 936 q^{82} - 438 q^{83} + 40 q^{84} + 126 q^{85} + 220 q^{86} + 120 q^{87} - 273 q^{89} - 156 q^{90} + 160 q^{91} + 756 q^{92} - 163 q^{93} - 288 q^{94} + 348 q^{95} - 32 q^{96} + 761 q^{97} + 486 q^{98}+O(q^{100}) \) 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 1.00000 4.00000 −3.00000 −2.00000 10.0000 −8.00000 −26.0000 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)

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 242.4.a.a 1
3.b odd 2 1 2178.4.a.r 1
4.b odd 2 1 1936.4.a.h 1
11.b odd 2 1 22.4.a.c 1
11.c even 5 4 242.4.c.k 4
11.d odd 10 4 242.4.c.d 4
33.d even 2 1 198.4.a.b 1
44.c even 2 1 176.4.a.c 1
55.d odd 2 1 550.4.a.e 1
55.e even 4 2 550.4.b.g 2
77.b even 2 1 1078.4.a.f 1
88.b odd 2 1 704.4.a.e 1
88.g even 2 1 704.4.a.g 1
132.d odd 2 1 1584.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.a.c 1 11.b odd 2 1
176.4.a.c 1 44.c even 2 1
198.4.a.b 1 33.d even 2 1
242.4.a.a 1 1.a even 1 1 trivial
242.4.c.d 4 11.d odd 10 4
242.4.c.k 4 11.c even 5 4
550.4.a.e 1 55.d odd 2 1
550.4.b.g 2 55.e even 4 2
704.4.a.e 1 88.b odd 2 1
704.4.a.g 1 88.g even 2 1
1078.4.a.f 1 77.b even 2 1
1584.4.a.k 1 132.d odd 2 1
1936.4.a.h 1 4.b odd 2 1
2178.4.a.r 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(242))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T - 10 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 16 \) Copy content Toggle raw display
$17$ \( T + 42 \) Copy content Toggle raw display
$19$ \( T + 116 \) Copy content Toggle raw display
$23$ \( T - 189 \) Copy content Toggle raw display
$29$ \( T - 120 \) Copy content Toggle raw display
$31$ \( T + 163 \) Copy content Toggle raw display
$37$ \( T + 409 \) Copy content Toggle raw display
$41$ \( T + 468 \) Copy content Toggle raw display
$43$ \( T + 110 \) Copy content Toggle raw display
$47$ \( T - 144 \) Copy content Toggle raw display
$53$ \( T - 90 \) Copy content Toggle raw display
$59$ \( T + 453 \) Copy content Toggle raw display
$61$ \( T + 20 \) Copy content Toggle raw display
$67$ \( T + 97 \) Copy content Toggle raw display
$71$ \( T + 465 \) Copy content Toggle raw display
$73$ \( T + 848 \) Copy content Toggle raw display
$79$ \( T - 742 \) Copy content Toggle raw display
$83$ \( T + 438 \) Copy content Toggle raw display
$89$ \( T + 273 \) Copy content Toggle raw display
$97$ \( T - 761 \) Copy content Toggle raw display
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