Properties

Label 1058.4.a.b
Level $1058$
Weight $4$
Character orbit 1058.a
Self dual yes
Analytic conductor $62.424$
Analytic rank $0$
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] = [1058,4,Mod(1,1058)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1058, 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("1058.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1058.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: \(62.4240207861\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 9 q^{3} + 4 q^{4} + 20 q^{5} - 18 q^{6} - 2 q^{7} + 8 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 9 q^{3} + 4 q^{4} + 20 q^{5} - 18 q^{6} - 2 q^{7} + 8 q^{8} + 54 q^{9} + 40 q^{10} + 52 q^{11} - 36 q^{12} + 43 q^{13} - 4 q^{14} - 180 q^{15} + 16 q^{16} + 50 q^{17} + 108 q^{18} + 74 q^{19} + 80 q^{20} + 18 q^{21} + 104 q^{22} - 72 q^{24} + 275 q^{25} + 86 q^{26} - 243 q^{27} - 8 q^{28} - 7 q^{29} - 360 q^{30} - 273 q^{31} + 32 q^{32} - 468 q^{33} + 100 q^{34} - 40 q^{35} + 216 q^{36} + 4 q^{37} + 148 q^{38} - 387 q^{39} + 160 q^{40} + 123 q^{41} + 36 q^{42} + 152 q^{43} + 208 q^{44} + 1080 q^{45} + 75 q^{47} - 144 q^{48} - 339 q^{49} + 550 q^{50} - 450 q^{51} + 172 q^{52} - 86 q^{53} - 486 q^{54} + 1040 q^{55} - 16 q^{56} - 666 q^{57} - 14 q^{58} - 444 q^{59} - 720 q^{60} - 262 q^{61} - 546 q^{62} - 108 q^{63} + 64 q^{64} + 860 q^{65} - 936 q^{66} - 764 q^{67} + 200 q^{68} - 80 q^{70} - 21 q^{71} + 432 q^{72} + 681 q^{73} + 8 q^{74} - 2475 q^{75} + 296 q^{76} - 104 q^{77} - 774 q^{78} - 426 q^{79} + 320 q^{80} + 729 q^{81} + 246 q^{82} - 902 q^{83} + 72 q^{84} + 1000 q^{85} + 304 q^{86} + 63 q^{87} + 416 q^{88} + 1272 q^{89} + 2160 q^{90} - 86 q^{91} + 2457 q^{93} + 150 q^{94} + 1480 q^{95} - 288 q^{96} + 342 q^{97} - 678 q^{98} + 2808 q^{99}+O(q^{100}) \) 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 −9.00000 4.00000 20.0000 −18.0000 −2.00000 8.00000 54.0000 40.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(23\) \( -1 \)

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 1058.4.a.b 1
23.b odd 2 1 46.4.a.b 1
69.c even 2 1 414.4.a.b 1
92.b even 2 1 368.4.a.e 1
115.c odd 2 1 1150.4.a.d 1
115.e even 4 2 1150.4.b.a 2
161.c even 2 1 2254.4.a.b 1
184.e odd 2 1 1472.4.a.j 1
184.h even 2 1 1472.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.b 1 23.b odd 2 1
368.4.a.e 1 92.b even 2 1
414.4.a.b 1 69.c even 2 1
1058.4.a.b 1 1.a even 1 1 trivial
1150.4.a.d 1 115.c odd 2 1
1150.4.b.a 2 115.e even 4 2
1472.4.a.a 1 184.h even 2 1
1472.4.a.j 1 184.e odd 2 1
2254.4.a.b 1 161.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):

\( T_{3} + 9 \) Copy content Toggle raw display
\( T_{5} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T - 20 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 52 \) Copy content Toggle raw display
$13$ \( T - 43 \) Copy content Toggle raw display
$17$ \( T - 50 \) Copy content Toggle raw display
$19$ \( T - 74 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 7 \) Copy content Toggle raw display
$31$ \( T + 273 \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T - 123 \) Copy content Toggle raw display
$43$ \( T - 152 \) Copy content Toggle raw display
$47$ \( T - 75 \) Copy content Toggle raw display
$53$ \( T + 86 \) Copy content Toggle raw display
$59$ \( T + 444 \) Copy content Toggle raw display
$61$ \( T + 262 \) Copy content Toggle raw display
$67$ \( T + 764 \) Copy content Toggle raw display
$71$ \( T + 21 \) Copy content Toggle raw display
$73$ \( T - 681 \) Copy content Toggle raw display
$79$ \( T + 426 \) Copy content Toggle raw display
$83$ \( T + 902 \) Copy content Toggle raw display
$89$ \( T - 1272 \) Copy content Toggle raw display
$97$ \( T - 342 \) Copy content Toggle raw display
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