Properties

Label 363.4.a.d
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: no (minimal twist has level 33)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} - 3 q^{6} + 26 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} - 3 q^{6} + 26 q^{7} - 15 q^{8} + 9 q^{9} - 4 q^{10} + 21 q^{12} + 32 q^{13} + 26 q^{14} + 12 q^{15} + 41 q^{16} - 74 q^{17} + 9 q^{18} + 60 q^{19} + 28 q^{20} - 78 q^{21} - 182 q^{23} + 45 q^{24} - 109 q^{25} + 32 q^{26} - 27 q^{27} - 182 q^{28} + 90 q^{29} + 12 q^{30} - 8 q^{31} + 161 q^{32} - 74 q^{34} - 104 q^{35} - 63 q^{36} - 66 q^{37} + 60 q^{38} - 96 q^{39} + 60 q^{40} - 422 q^{41} - 78 q^{42} - 408 q^{43} - 36 q^{45} - 182 q^{46} - 506 q^{47} - 123 q^{48} + 333 q^{49} - 109 q^{50} + 222 q^{51} - 224 q^{52} + 348 q^{53} - 27 q^{54} - 390 q^{56} - 180 q^{57} + 90 q^{58} - 200 q^{59} - 84 q^{60} - 132 q^{61} - 8 q^{62} + 234 q^{63} - 167 q^{64} - 128 q^{65} - 1036 q^{67} + 518 q^{68} + 546 q^{69} - 104 q^{70} + 762 q^{71} - 135 q^{72} + 542 q^{73} - 66 q^{74} + 327 q^{75} - 420 q^{76} - 96 q^{78} + 550 q^{79} - 164 q^{80} + 81 q^{81} - 422 q^{82} + 132 q^{83} + 546 q^{84} + 296 q^{85} - 408 q^{86} - 270 q^{87} + 570 q^{89} - 36 q^{90} + 832 q^{91} + 1274 q^{92} + 24 q^{93} - 506 q^{94} - 240 q^{95} - 483 q^{96} + 14 q^{97} + 333 q^{98}+O(q^{100}) \) 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
1.00000 −3.00000 −7.00000 −4.00000 −3.00000 26.0000 −15.0000 9.00000 −4.00000
\(n\): 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.d 1
3.b odd 2 1 1089.4.a.e 1
11.b odd 2 1 33.4.a.b 1
33.d even 2 1 99.4.a.a 1
44.c even 2 1 528.4.a.h 1
55.d odd 2 1 825.4.a.f 1
55.e even 4 2 825.4.c.f 2
77.b even 2 1 1617.4.a.d 1
88.b odd 2 1 2112.4.a.u 1
88.g even 2 1 2112.4.a.h 1
132.d odd 2 1 1584.4.a.l 1
165.d even 2 1 2475.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 11.b odd 2 1
99.4.a.a 1 33.d even 2 1
363.4.a.d 1 1.a even 1 1 trivial
528.4.a.h 1 44.c even 2 1
825.4.a.f 1 55.d odd 2 1
825.4.c.f 2 55.e even 4 2
1089.4.a.e 1 3.b odd 2 1
1584.4.a.l 1 132.d odd 2 1
1617.4.a.d 1 77.b even 2 1
2112.4.a.h 1 88.g even 2 1
2112.4.a.u 1 88.b odd 2 1
2475.4.a.e 1 165.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T - 26 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 32 \) Copy content Toggle raw display
$17$ \( T + 74 \) Copy content Toggle raw display
$19$ \( T - 60 \) Copy content Toggle raw display
$23$ \( T + 182 \) Copy content Toggle raw display
$29$ \( T - 90 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 66 \) Copy content Toggle raw display
$41$ \( T + 422 \) Copy content Toggle raw display
$43$ \( T + 408 \) Copy content Toggle raw display
$47$ \( T + 506 \) Copy content Toggle raw display
$53$ \( T - 348 \) Copy content Toggle raw display
$59$ \( T + 200 \) Copy content Toggle raw display
$61$ \( T + 132 \) Copy content Toggle raw display
$67$ \( T + 1036 \) Copy content Toggle raw display
$71$ \( T - 762 \) Copy content Toggle raw display
$73$ \( T - 542 \) Copy content Toggle raw display
$79$ \( T - 550 \) Copy content Toggle raw display
$83$ \( T - 132 \) Copy content Toggle raw display
$89$ \( T - 570 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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