Properties

Label 363.4.a.g
Level $363$
Weight $4$
Character orbit 363.a
Self dual yes
Analytic conductor $21.418$
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] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, 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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 13 q^{5} - 12 q^{6} + 26 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 13 q^{5} - 12 q^{6} + 26 q^{7} + 9 q^{9} - 52 q^{10} - 24 q^{12} - 73 q^{13} + 104 q^{14} + 39 q^{15} - 64 q^{16} + 31 q^{17} + 36 q^{18} - 108 q^{19} - 104 q^{20} - 78 q^{21} - 86 q^{23} + 44 q^{25} - 292 q^{26} - 27 q^{27} + 208 q^{28} - 207 q^{29} + 156 q^{30} + 208 q^{31} - 256 q^{32} + 124 q^{34} - 338 q^{35} + 72 q^{36} + 45 q^{37} - 432 q^{38} + 219 q^{39} + 247 q^{41} - 312 q^{42} - 450 q^{43} - 117 q^{45} - 344 q^{46} - 500 q^{47} + 192 q^{48} + 333 q^{49} + 176 q^{50} - 93 q^{51} - 584 q^{52} - 441 q^{53} - 108 q^{54} + 324 q^{57} - 828 q^{58} + 598 q^{59} + 312 q^{60} + 378 q^{61} + 832 q^{62} + 234 q^{63} - 512 q^{64} + 949 q^{65} + 494 q^{67} + 248 q^{68} + 258 q^{69} - 1352 q^{70} - 594 q^{71} + 1034 q^{73} + 180 q^{74} - 132 q^{75} - 864 q^{76} + 876 q^{78} + 352 q^{79} + 832 q^{80} + 81 q^{81} + 988 q^{82} + 360 q^{83} - 624 q^{84} - 403 q^{85} - 1800 q^{86} + 621 q^{87} - 351 q^{89} - 468 q^{90} - 1898 q^{91} - 688 q^{92} - 624 q^{93} - 2000 q^{94} + 1404 q^{95} + 768 q^{96} + 1079 q^{97} + 1332 q^{98}+O(q^{100}) \) 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
4.00000 −3.00000 8.00000 −13.0000 −12.0000 26.0000 0 9.00000 −52.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

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 363.4.a.g yes 1
3.b odd 2 1 1089.4.a.b 1
11.b odd 2 1 363.4.a.a 1
33.d even 2 1 1089.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.a 1 11.b odd 2 1
363.4.a.g yes 1 1.a even 1 1 trivial
1089.4.a.b 1 3.b odd 2 1
1089.4.a.j 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2} - 4 \) Copy content Toggle raw display
\( T_{5} + 13 \) Copy content Toggle raw display
\( T_{7} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 13 \) Copy content Toggle raw display
$7$ \( T - 26 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 73 \) Copy content Toggle raw display
$17$ \( T - 31 \) Copy content Toggle raw display
$19$ \( T + 108 \) Copy content Toggle raw display
$23$ \( T + 86 \) Copy content Toggle raw display
$29$ \( T + 207 \) Copy content Toggle raw display
$31$ \( T - 208 \) Copy content Toggle raw display
$37$ \( T - 45 \) Copy content Toggle raw display
$41$ \( T - 247 \) Copy content Toggle raw display
$43$ \( T + 450 \) Copy content Toggle raw display
$47$ \( T + 500 \) Copy content Toggle raw display
$53$ \( T + 441 \) Copy content Toggle raw display
$59$ \( T - 598 \) Copy content Toggle raw display
$61$ \( T - 378 \) Copy content Toggle raw display
$67$ \( T - 494 \) Copy content Toggle raw display
$71$ \( T + 594 \) Copy content Toggle raw display
$73$ \( T - 1034 \) Copy content Toggle raw display
$79$ \( T - 352 \) Copy content Toggle raw display
$83$ \( T - 360 \) Copy content Toggle raw display
$89$ \( T + 351 \) Copy content Toggle raw display
$97$ \( T - 1079 \) Copy content Toggle raw display
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