Properties

Label 363.4.a.c
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)
 
\(f(q)\) \(=\) \( q - q^{2} - 3 q^{3} - 7 q^{4} + 7 q^{5} + 3 q^{6} - 4 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 3 q^{3} - 7 q^{4} + 7 q^{5} + 3 q^{6} - 4 q^{7} + 15 q^{8} + 9 q^{9} - 7 q^{10} + 21 q^{12} - 43 q^{13} + 4 q^{14} - 21 q^{15} + 41 q^{16} + 41 q^{17} - 9 q^{18} + 72 q^{19} - 49 q^{20} + 12 q^{21} + 104 q^{23} - 45 q^{24} - 76 q^{25} + 43 q^{26} - 27 q^{27} + 28 q^{28} + 273 q^{29} + 21 q^{30} - 272 q^{31} - 161 q^{32} - 41 q^{34} - 28 q^{35} - 63 q^{36} - 165 q^{37} - 72 q^{38} + 129 q^{39} + 105 q^{40} - 403 q^{41} - 12 q^{42} - 120 q^{43} + 63 q^{45} - 104 q^{46} - 220 q^{47} - 123 q^{48} - 327 q^{49} + 76 q^{50} - 123 q^{51} + 301 q^{52} - 741 q^{53} + 27 q^{54} - 60 q^{56} - 216 q^{57} - 273 q^{58} - 112 q^{59} + 147 q^{60} + 858 q^{61} + 272 q^{62} - 36 q^{63} - 167 q^{64} - 301 q^{65} + 284 q^{67} - 287 q^{68} - 312 q^{69} + 28 q^{70} - 624 q^{71} + 135 q^{72} - 586 q^{73} + 165 q^{74} + 228 q^{75} - 504 q^{76} - 129 q^{78} - 308 q^{79} + 287 q^{80} + 81 q^{81} + 403 q^{82} - 84 q^{84} + 287 q^{85} + 120 q^{86} - 819 q^{87} - 321 q^{89} - 63 q^{90} + 172 q^{91} - 728 q^{92} + 816 q^{93} + 220 q^{94} + 504 q^{95} + 483 q^{96} + 179 q^{97} + 327 q^{98}+O(q^{100}) \) 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 7.00000 3.00000 −4.00000 15.0000 9.00000 −7.00000
\(n\): 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.c 1
3.b odd 2 1 1089.4.a.h 1
11.b odd 2 1 363.4.a.e yes 1
33.d even 2 1 1089.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.c 1 1.a even 1 1 trivial
363.4.a.e yes 1 11.b odd 2 1
1089.4.a.d 1 33.d even 2 1
1089.4.a.h 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 7 \) Copy content Toggle raw display
\( T_{7} + 4 \) 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
$5$ \( T - 7 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 43 \) Copy content Toggle raw display
$17$ \( T - 41 \) Copy content Toggle raw display
$19$ \( T - 72 \) Copy content Toggle raw display
$23$ \( T - 104 \) Copy content Toggle raw display
$29$ \( T - 273 \) Copy content Toggle raw display
$31$ \( T + 272 \) Copy content Toggle raw display
$37$ \( T + 165 \) Copy content Toggle raw display
$41$ \( T + 403 \) Copy content Toggle raw display
$43$ \( T + 120 \) Copy content Toggle raw display
$47$ \( T + 220 \) Copy content Toggle raw display
$53$ \( T + 741 \) Copy content Toggle raw display
$59$ \( T + 112 \) Copy content Toggle raw display
$61$ \( T - 858 \) Copy content Toggle raw display
$67$ \( T - 284 \) Copy content Toggle raw display
$71$ \( T + 624 \) Copy content Toggle raw display
$73$ \( T + 586 \) Copy content Toggle raw display
$79$ \( T + 308 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 321 \) Copy content Toggle raw display
$97$ \( T - 179 \) Copy content Toggle raw display
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